Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.46.0-wmf.23 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk Portal:Foreign Language Learning 102 3638 2803467 2799730 2026-04-08T03:16:54Z CarlessParking 3064444 2803467 wikitext text/x-wiki <div style="margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;"> {{center top}}'''[[Portal:Humanities|F<small>ACULTY FOR</small> H<small>UMANITIES</small>]]'''{{center bottom}} [[School:Art and Design|Art and Design]] · [[School:Classics|Classics]] · [[Portal:Mythology|Mythology]] · [[School:Law|Law]] · [[School:Language and Literature|Language and Literature]] · [[School:Music and Dance|Music]] · [[School:Philosophy|Philosophy]] · [[School:Theology|Theology]]</div> {{center top}} {|style="width:100%;margin-top:+.7em;background-color:#fcfcfc; {{text color default}};border:1px solid #ccc" |style="width:50%;color:#000"| {|style="width:280px;border:solid 0px;background:none; color:inherit;" |- |style="width:280px;text-align:center;white-space:nowrap;color:#000" | <div style="font-size:133%;border:none;margin: 0;padding:.1em;color:#000">'''W<small>ELCOME TO THE </small> C<small>ENTRE FOR</small><br>F<small>OREIGN</small> L<small>ANGUAGE</small> L<small>EARNING</small>,'''</div> <div style="top:+0.2em;font-size: 95%">part of the School of [[School:Language and Literature|language and literature]].</div> <div style="width:100%;text-align:center;font-size:80%;">You may also be interested in [[Portal:Practical Arts and Sciences|Practical arts and sciences]].</div> |} |style="width:18%;font-size:95%;color:#000"| *[[Literary studies]] *[[Portal:Foreign Language Learning/Participant Coordination|Sign up!]] |style="width:10%;font-size:95%";color:#000"| [[Image:Nuvola apps bookcase.svg|center|50px|Language and Literature]] |} {{center bottom}} <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Foreign Language Learning</h2 > [[Image:Globe of letters.svg|right|88px|Languages]] Hello and welcome to the Wikiversity Centre for Foreign Language Learning. Here you may learn foreign languages and explore their cultures, as well as teach others the languages that you speak. Perhaps you are also interested in the Wikiversity's schools of various [[Portal:Ethnic Studies|ethnic studies]]. The study of the structure of a language and its development is called ''[[w:linguistics|linguistics]]''. Wikiversity's [[School:Linguistics|School of Linguistics]] is part of the [[Portal:Social Sciences|Faculty of Social Sciences]]. '''Note:''' Participants interested in language projects not yet featured here can coordinate with others at the Center's [[Portal:Foreign Language Learning/Participant Coordination|Participant Coordination]] page. </div> <div style="display:block;width:99%;float:left"> <div style="width:53%;display:block;float:left;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Indo-European languages</h2 > [[Image:Indo-European_branches_map.svg|right|300px|]] * '''Germanic languages''' ** [[Portal:English Language|English]] ** [[Afrikaans|Afrikaans]] ** [[Portal:Dutch|Dutch]] ** [[Portal:German|German]] ** [[Portal:Danish|Danish]] ** [[Portal:Swedish|Swedish]] ** Norwegian ***[[Portal:Nynorsk|Nynorsk]] ***[[Portal:Norwegian|Bokmål]] ** [[Portal:Icelandic|Icelandic]] ** [[Portal:Faroese|Faroese]] * '''Italic languages''' ** [[Portal:Latin|Latin]] ** '''Romance languages''' *** [[Portal:Catalan|Catalan]] *** [[Portal:French|French]] **** [[Portal:Canadian French|Canadian French]] *** [[Portal:Italian|Italian]] *** [[Portal:Occitan|Occitan]] *** [[Portal:Picard|Picard]] *** [[Portal:Portuguese|Portuguese]] *** [[Portal:Romansh|Romansh]] *** [[Portal:Romanian|Romanian]] *** [[Portal:Sardinian|Sardinian]] **** [[Portal:Logudorese|Logudorese]] ***** [[Portal:Nuorese|Nuorese]] **** [[Portal:Campidanese|Campidanese]] *** [[Portal:Spanish|Spanish]] * '''Slavic languages''' ** [[Portal:Czech|Czech]] ** [[Portal:Slovak|Slovak]] ** [[Portal:Polish|Polish]] ** [[Topic:Russian|Russian]] ** [[Portal:Ukrainian|Ukrainian]] ** [[Portal:Belarusian|Belarusian]] ** [[Portal:Slovene|Slovene]] ** [[Portal:Serbo-Croatian|Serbo-Croatian]] (Croatian, Serbian, Bosnian) ** [[Portal:Macedonian|Macedonian]] ** [[Portal:Bulgarian|Bulgarian]] * '''Baltic languages''' ** [[Portal:Latvian|Latvian]] ** [[Portal:Lithuanian|Lithuanian]] * '''Celtic languages''' ** [[Portal:Irish|Irish]] ** [[Portal:Scottish Gaelic|Scottish Gaelic]] ** [[Portal:Brythonic Celtic Languages Division|Brythonic Celtic Languages Division]] *** [[Portal:Breton|Breton]] *** [[Portal:Cornish|Cornish]] *** [[Welsh|Welsh]] * '''Indo-Iranian languages''' ** '''Indo-Aryan languages''' *** '''Hindi languages''' **** [[Portal:Hindustani|Hindustani]] ***** [[Portal:Hindi|Hindi]] ***** [[Portal:Urdu|Urdu]] **** [[Portal:Awadhi|Awadhi]] ***** [[Fiji Hindi]] *** [[Bengali|Bengali]] *** [[Portal:Punjabi|Punjabi]] *** [[Portal:Marathi|Marathi]] *** [[Portal:Gujarati|Gujarati]] *** [[Portal:Maithili|Maithili]] *** [[Portal:Oriya|Oriya]] *** [[Portal:Sindhi|Sindhi]] *** [[Portal:Nepali|Nepali]] *** [[Portal:Sinhala|Sinhala]] *** [[Portal:Assamese|Assamese]] ** '''Iranian languages''' *** [[Portal:Persian|Farsi]] * '''Hellenic languages''' ** [[Introductory Ancient Greek Language|Ancient Greek]] ** [[Portal:Greek|Greek]] * '''Other languages''' ** [[Portal:Albanian|Albanian]] *** [[Standard Albanian|Standard (Tosk)]] *** [[Gheg Albanian|Gheg]] ** [[Portal:Armenian|Armenian]] :'''[[w:Indo-European languages|read more...]]''' </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Languages of the Americas</h2 > [[Image:P history.png|right|55px|]] *[[:Category:Languages of North America|Languages of North America]] **'''Iroquoian languages''' *** [[Portal:Cherokee|Cherokee]] **'''Na-Dené languages''' *** [[Portal:Navajo|Navajo]] *[[:Category:Ancient languages of central america|Languages of Central America]] *[[:Category:Languages of South America|Languages of South America]] ** [[Portal:Quechua|Quechua]] :'''[[w:Indigenous languages of the Americas|read more...]]''' </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Semitic and African languages</h2 > [[Image:Nuvola apps edu languages.svg|right|44px|]] *'''Afro-Asiatic languages''' **'''Berber languages''' *** [[Portal:Tuareg|Tuareg]] *** [[Portal:Kabyle|Kabyle]] **'''Chadic languages''' *** [[Portal:Hausa|Hausa]] **'''Cushitic languages''' *** [[Portal:Oromo|Oromo]] *** [[Portal:Somali|Somali]] **'''Egyptian languages''' *** [[Portal:Ancient Egyptian|Ancient Egyptian]] *** [[Portal:Coptic|Coptic]] **'''Semitic languages''' *** Arabic languages **** [[Portal:Arabic|Modern Standard Arabic]] **** [[Portal:Egyptian Arabic|Egyptian Arabic]] **** [[Portal:Moroccan Arabic|Moroccan Arabic]] **** [[Portal:Maltese|Maltese]] *** [[Portal:Hebrew|Hebrew]] **** [[Portal:Modern Hebrew|Modern Hebrew]] **** [[Portal:Biblical Hebrew|Biblical Hebrew]] *** [[Portal:Amharic|Amharic]] **'''Omotic languages''' :'''[[w:Afro-Asiatic languages|read more...]]''' *'''Nilo-Saharan languages''' **'''Eastern Sudanic languages''' *** [[Portal:Luo|Luo]] *** [[Portal:Maasai|Maasai]] :'''[[w:Nilo-Saharan languages|read more...]]''' *'''Niger–Congo / Bantu languages''' ** [[Portal:Luganda|Luganda]] ** [[Portal:Sesotho|Sesotho]] ** [[Portal:Swahili|Swahili]] :'''[[w:Niger–Congo languages|read more...]]''' </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Austronesian languages</h2 > * [[Portal:Bikol|Bikol]] * [[Portal:Malay|Malay]] ** [[Southeast Asian Languages/Bahasa Malaysia|Malaysian]] ** [[Southeast Asian Languages/Bahasa Indonesia|Indonesian]] * [[Portal:Tagalog|Tagalog]] ** [[Southeast Asian Languages/Philippine Languages/Filipino One|Filipino]] * [[Portal:Javanese|Javanese]] * [[Portal:Sundanese|Sundanese]] * [[Portal:Cebuano|Cebuano]] * Malagasy ** [[Portal:Northern Betsimisaraka Malagasy|Northern Betsimisaraka Malagasy]] ** [[Portal:Plateau Malagasy|Plateau Malagasy]] ** [[Portal:Tandroy-Mahafaly Malagasy|Tandroy-Mahafaly Malagasy]] ** [[Portal:Tsimihety Malagasy|Tsimihety Malagasy]] ** [[Portal:Southern Betsimisaraka Malagasy|Southern Betsimisaraka Malagasy]] * [[Portal:Maori|Maori]] * [[Portal:Hawaiian|Hawaiian]] :'''[[w:Austronesian languages|read more...]]''' </div> </div > <div style="width:43%;display:block;float:right;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Ural–Altaic languages</h2 > [[Image:Fenno-Ugrian languages.png|right|150px|]] *'''Uralic languages''' **'''Finnic languages''' *** [[Portal:Finnish|Finnish]] *** [[Portal:Estonian|Estonian]] *** [[Portal:Karelian|Karelian]] *** [[Portal:Votic|Votic]] **'''Ugric languages''' *** [[Portal:Hungarian|Hungarian]] :'''[[w:Uralic languages|read more...]]''' [[Image:Lenguas altaicas.png|right|150px|]] *'''Turkic languages''' ** [[Turkish]] ** [[Portal:Azerbaijani|Azerbaijani]] ** [[Portal:Uzbek|Uzbek]] ** [[Portal:Kazakh|Kazakh]] :'''[[w:Turkic languages|read more...]]''' *'''Mongolic languages''' ** [[Portal:Mongolian|Mongolian]] :'''[[w:Mongolic languages|read more...]]''' *'''Tungusic languages''' ** [[Portal:Manchu|Manchu]] :'''[[w:Tungusic languages|read more...]]''' * '''Japonic languages''' ** [[Portal:Japanese|Japanese]] ** '''Ryukyuan languages''' *** [[Portal:Okinawan|Okinawan]] :'''[[w:Japonic languages|read more...]]''' * [[Portal:Korean|Korean]] :'''[[w:Koreanic languages|read more...]]''' </div > <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">East and South Asian languages</h2 > '''East Asian languages''' [[Image:Sino-tibetan languages - branches.png|right|150px|]] *'''Sino-Tibetan languages''' ** [[Portal:Tibetan|Tibetan]] ** [[Portal:Chinese|Chinese]] languages ***[[Mandarin]] ***[[Cantonese]] ** [[Portal:Burmese|Burmese]] :'''[[w:Sino-Tibetan languages|read more...]]''' '''[[Portal:Southeast Asian languages|Southeast Asian languages]]''' [[Image:Austroasiatic-en.svg|right|150px|]] *'''Austro-Asiatic languages''' ** [[Portal:Vietnamese|Vietnamese]] ** [[Portal:Khmer|Khmer]] ** [[Potal:Santali|Santali]] ** [[Portal:Ho|Ho]] ** [[Portal:Mundari|Mundari]] ** [[Portal:Khasi|Khasi]] :'''[[w:Austro-Asiatic languages|read more...]]''' [[Image:Hmong-Mien-en.svg|right|150px|]] *'''Hmong–Mien languages''' ** Hmongic ('''[[w:Hmongic languages|read more...]]''') ** Mienic ('''[[w:Mienic languages|read more...]]''') :'''[[w:Hmong–Mien languages|read more...]]''' [[Image:Taikadai-en.svg|right|150px|]] *'''Tai–Kadai languages''' ** [[Portal:Lao|Lao]] ** [[Portal:Thai|Thai]] :'''[[w:Tai–Kadai languages|read more...]]''' '''South Asian languages''' [[Image:Dravidian subgroups.png|right|150px|]] *'''Dravidian languages''' ** [[Portal:Telugu|Telugu]] ** [[Portal:Tamil|Tamil]] ** [[Portal:Kannada|Kannada]] ** [[Portal:Malayalam|Malayalam]] ** [[Portal:Brahui|Brahui]] ** [[Portal:Kurux|Kurux]] ** [[Portal:Gondi|Gondi]] ** [[Portal:Tulu|Tulu]] :'''[[w:Dravidian languages|read more...]]''' '''[[w:Languages of Asia|read more...]]''' </div > <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Other languages and language isolates</h2 > [[Image:Nuvola apps kfind.png|right|56px|]] * [[Topic:Basque|Basque]] *'''Kartvelian languages''' ** [[Topic:Georgian|Georgian]] ** [[Topic:Mingrelian|Mingrelian]] ** [[Topic:Svan|Svan]] ** [[Topic:Laz|Laz]] :'''[[w:Kartvelian languages|read more...]]''' *'''Northwest Caucasian languages''' ** [[Topic:Akbhaz|Abkhaz]] ** [[Topic:Adyghe|Adyghe]] :'''[[w:Northwest Caucasian languages|read more...]]''' *'''Northeast Caucasian languages''' ** [[Portal:Chechen|Chechen]] ** [[Portal:Ingush|Ingush]] ** [[Portal:Avar|Avar]] ** [[Portal:Dargwa|Dargwa]] ** [[Portal:Lak|Lak]] ** [[Portal:Tabasaran|Tabasaran]] :'''[[w:Northeast Caucasian languages|read more...]]''' * [[Portal:Hadza|Hadza]] * [[Portal:Sandawe|Sandawe]] </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Constructed languages</h2 > * [[Esperanto]] * [[Ido]] * [[Interlingua|Interlingua]] * [[Ithkuil]] * [[Portal:Klingon|Klingon]] * [[Portal:Láadan|Láadan]] * [[Portal:Loglan|Loglan]] * [[Portal:Lojban|Lojban]] * [[Portal:Novial|Novial]] * [[Portal:Slovio|Slovio]] * [[Turoke|Turoke]] * [[Portal:Unilingua|Unilingua]] * [[Portal:Volapük|Volapük]] *'''Tolkienesque languages''' ** [[Quenya]] ** [[Sindarin]] ** [[Portal:Tolkien Languages|other]] :'''[[w:List of constructed languages|read more...]]''' </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Sign languages</h2 > * [[Portal:American Sign Language|American Sign Language]] * [[Portal:French Sign Language|French Sign Language]] * [[Portal:Chinese Sign Language|Chinese Sign Language]] * [[Portal:Japanese Sign Language|Japanese Sign Language]] * [[Portal:Dutch Sign Language|Dutch Sign Language]] :'''[[w:List of sign languages|read more...]]''' </div > </div > </div > <div style="display:block;float:left;width:100%;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Useful textbooks from [[b:|Wikibooks]]</h2 > {| |style="width:44%;font-size:95%;color:#000"| *[[b:Wikibooks:Language and literature bookshelf|Books about language and literature]] *[[b:Wikibooks:Languages bookshelf|Textbooks for learning languages]] |style="width:44%;font-size:95%"| *[[b:Wikibooks:Humanities bookshelf|Humanities bookshelf]] *[[b:Wikibooks:Social sciences bookshelf|Social sciences bookshelf]] |style="width:11%;font-size:95%"| [[Image:Nuvola apps bookcase.svg|center|50px|Reading]] |} </div > </div > <div style="display:block;float:left;width:100%;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top; background-color:#f9f9ff; {{text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Associated Wikiversity schools and faculties</h2 > *[[School:Language and Literature|The School of Language and Literature]] [[School:Language and Literature/Resource catalogue|resource catalogue]] *[[Portal:Ethnic Studies|List of various schools of ethnic studies]] *[[School:Linguistics|School of Linguistics]] </div > </div > __NOEDITSECTION__ __NOTOC__ [[Category:Foreign Language Learning| ]] [[Category:Languages]] [[Category:Interlingual Beta Club]] [[Category:Languages and Language families]] [[Category:Multilingual Studies]] [[fr:Faculté:Langues étrangères]] rqbz605lswz2gxptfr5ollt68ihnog1 Portal:Translation 102 7995 2803472 1861873 2026-04-08T04:38:23Z CarlessParking 3064444 /* Languages */ 2803472 wikitext text/x-wiki [[Portal:Translation|Translation Department]] • <small>[[Portal:Multilingual Studies|Wikiversity Institute for Multilingual Studies]]</small> ==Subdivision news== * '''23 October 2006''' - Subdivision founded! * '''29 November 2006''' - Setting up at [http://beta.wikiversity.org/wiki/Translation beta.wikiversity] * ... == Learning Groups == See [[Portal:Multilingual Studies|Topic:Multilingual Studies]] for a more in-depth listing for Learning groups and topics within the larger [[Multilingual worksheet|Multilingulism Practicum]]. This is just a rough outline: [[School:Language and Literature]]: *[[Portal:Foreign Language Learning|Topic:Foreign Language Learning]] * ... [[School:Linguistics]]: *[[Topic:History of Linguistics|History of Linguistics]] *[[Portal:English Language|Topic:English Language]] *... [[School:Media studies|School:Media Studies]] *[[Wikiversity the Movie]] *[[Wikiversity the Movie/Multi-lingual|Wikiversity the Movie/multi-lingual]] *... [[School:Computer Science]] *[[Portal:Computational linguistics|Topic:Computational linguistics]] *[[Computer-assisted translation]] *... == Learning materials == [[Wikiversity:Translator's Handbook]]. * Introduction ''See [[Wikiversity:Essay Contest]]'' * [[Portal:Languages|Languages]] ''Will discuss [[:Category:Languages and Language families]] in detail covering'' * [[Topic:Linguistics]] * [[Portal:Computational linguistics|Topic:Lexicography]] * [[Portal:Translation|Appendix]] == Multilingualism at Wikiversity == [[Wikiversity:Multilingualism]] will be a centralized local resource for the English version of Wikiversity (en.wikiversity.org) ''about'' other versions of Wikiversity and beta projects. [[Wikiversity translations]]: *[[Wikiversity:Main Page|English]] *[[:de:Hauptseite]] [http://de.wikiversity.org Hauptseite] *[[:es:Portada]] [http://es.wikiversity.org Portada] Links to betas: *... == Wikiversity Translators == {{tl|translator}} @{{tl|translators}} ... {{translator}} <br><br> {{translators}} [[:Category:Wikiversity Translators]] {{construct}} == Wikimedia Service community == The following sections cover translation-related topics and list [[Wikiversity:Service|Translation services]] by and for the Wikimedia-Wikiversity [[Wikiversity:Service community]]. === Meta === [[m:Main Page|Meta]] has a cornucopia of resources for translators! [[m:Multilingualism]]: *[[m:Translators]] *[[m:Translation process]] *[[m:Interlingual coordination]] *[[m:Meta:Conventions for multilingualism]] *[[m:Wikipedia Machine Translation Project]] *... [[m:Category:Multilingualism]] *[[m:Category:Transbabel]] *[[m:Category:Translator's Templates]] *... <small>Note: We have adapted the meta Tranlator's Templates and are working on an adaptation of the [[Transbabel Multilingual Schema]] here at [[Wikiversity:Templates#Translation]] See [[Template:Translators]].</small> === Wikipedia === [[w:Main Page|Wikipedia]] ==== Articles ==== '''[[w:Meaning (linguistics)|Meanings]]:''' [[w:grammar|grammar]], [[w:semantics|semantics]], [[w:syntax|syntax]], [[w:idiom|idiom]], [[w:word|word]], [[w:phrase|phrase]], [[w:sentance|sentance]], [[w:source text|source text]], [[w:source language|source language]], [[w:target language|target language]], ... '''[[w:Language|Language]]:''' [[w:Natural language|Natural language]], [[w:Language family|Language family]], [[w:Linguistic typology|Linguistic typology]], [[w:Writing systems|Writing systems]], [[w:List of languages|List of languages]], [[w:List of official languages|List of official languages]], [[w:List of common phrases in various languages|List of common phrases in various languages]], ... '''[[w:Linguistics|Linguistics]]:''' [[w:Morphology (linguistics)|Morphology]], [[w:Etymology|Etymology]], [[w:Corpus linguistics|Corpus linguistics]], [[w:Text corpus|Text corpus]], [[w:Cognitive linguistics|Cognitive linguistics]], [[w:Computational linguistics|Computational linguistics]], [[w:Machine translation|Machine translation]], [[w:Speech recognition|Speech recognition]], [[w:Language recognition chart|Language recognition chart]], [[w:Part-of-speech tagging|Part-of-speech tagging]], ... '''[[w:Translation|Translation]]:''' [[w:translation dictionary|translation dictionary]], [[w:Translation process|Translation process]], [[w:Translation studies|Translation studies]], [[w:Translation unit|Translation unit]], [[w:Computer-assisted translation|Computer-assisted translation]], [[w:Machine translation|Machine translation]], ... ==== Projects ==== '''[[w:Wikipedia:Wikipedia Embassy|Wikipedia Embassy]]:''' *[[w:Wikipedia:Multilingual coordination|Wikipedia:Multilingual coordination]] *[[w:Wikipedia:Wikipedians]] *[[w:Wikipedia:Interwikimedia link]] *[[w:Wikipedia:Interlanguage links]] *... '''[[w:Wikipedia:WikiProject Linguistics|Wikipedia:WikiProject Linguistics]]:''' *[[w:Wikipedia:WikiProject Languages|WikiProject Languages]] *[[w:Wikipedia:WikiProject Language families|WikiProject Language families]] *[[w:Wikipedia:WikiProject Writing systems|WikiProject Writing systems]] *[[w:Wikipedia:WikiProject Latin alphabet|WikiProject Latin alphabet]] *[[w:Wikipedia:WikiProject Phonetics|WikiProject Phonetics]] *[[w:Wikipedia:WikiProject Constructed languages|WikiProject Constructed languages]] *[[w:Wikipedia:WikiProject Theoretical Linguistics|WikiProject Theoretical Linguistics]] '''[[w:Wikipedia:WikiProject Computer science|Wikipedia:WikiProject Computer science]]''' ==== Languages ==== List of Wikipedias in other languages: #[http://ab.wikipedia.org <span lang="ab-Cyrl">аҧсуа бысжѡа</span> (Abkhazian)] #[http://af.wikipedia.org Afrikaans] #[http://als.wikipedia.org Alemannisch (Alemannic)] #[http://am.wikipedia.org <span lang="am" dir="rtl">አማርኛ</span> (Amharic)] #[http://ar.wikipedia.org <span lang="ar" dir="rtl">‫العربية‬</span> (Arabic)] #[http://an.wikipedia.org Aragonés (Aragonese)] #[http://roa-rup.wikipedia.org Armâneashti (Aromanian)] #[http://as.wikipedia.org <span lang="as">অসমীয়া</span>&nbsp;/ <span lang="as-Latn">Asami</span> (Assamese)] #[http://arc.wikipedia.org <span lang="arc">ܣܘܪܬ</span> (Assyrian Neo-Aramaic)] #[http://ast.wikipedia.org Asturianu (Asturian)] #[http://gn.wikipedia.org Avañe'ẽ (Guarani)] #[http://av.wikipedia.org Авар (Avar)] #[http://ay.wikipedia.org Aymar (Aymara)] #[http://az.wikipedia.org <span lang="az">Azərbaycan dili</span>&nbsp;/ <span lang="az-Arab" dir="rtl">آذربايجان ديلی</span> (Azerbaijani)] #[http://id.wikipedia.org Bahasa Indonesia (Indonesian)] #[http://ms.wikipedia.org Bahasa Melayu (Malay)] #[http://bm.wikipedia.org Bamanankan (Bambara)] #[http://bn.wikipedia.org <span lang="bn">বাংলা</span> (Bengali)] #[http://zh-min-nan.wikipedia.org Bân-lâm-gú (Southern Min)] #[http://jv.wikipedia.org Basa Jawa (Javanese)] #[http://map-bms.wikipedia.org Basa Banyumasan (Banyumasan)] #[http://su.wikipedia.org Basa Sunda (Sundanese)] #[http://ba.wikipedia.org <span lang="ba">Башҡорт</span> (Bashkir)] #[http://be.wikipedia.org <span lang="be">Беларуская</span> (Belarusian)] #[http://bh.wikipedia.org <span lang="bh">भोजपुरी</span> (Bihari)] #[http://mt.wikipedia.org bil-Malti (Maltese)] #[http://bi.wikipedia.org Bislama] #[http://bo.wikipedia.org <span lang="bo">བོད་ཡིག</span>&nbsp;/ <span lang="bo-Latn">Bod skad</span> (Tibetan)] #[http://bs.wikipedia.org Bosanski (Bosnian)] #[http://br.wikipedia.org Brezhoneg (Breton)] #[http://bug.wikipedia.org Basa Ugi (Buginese)] #[http://bg.wikipedia.org <span lang="bg-Cyrl">Български</span> (Bulgarian)] #[http://ca.wikipedia.org Català (Catalan)] #[http://bcl.wikipedia.org Central Bikol (Bikol Central)] #[http://cbk-zam.wikipedia.org Chavacano (Chavacano de Zamboanga)] #[http://cho.wikipedia.org Cahta (Choctaw)] #[http://ch.wikipedia.org Chamorru (Chamorro)] #[http://cv.wikipedia.org <span lang="cv">Чӑваш</span> (Chuvash)] #[http://cs.wikipedia.org Čeština (Czech)] #[http://ny.wikipedia.org chiChewa] #[http://sn.wikipedia.org chiShona (Shona)] #[http://tum.wikipedia.org chiTumbuka (Tumbuka)] #[http://ve.wikipedia.org Chivenda (Venda)] #[http://co.wikipedia.org Corsu (Corsican)] #[http://za.wikipedia.org Cuengh (Zhuang)] #[http://cy.wikipedia.org Cymraeg (Welsh)] #[http://da.wikipedia.org Dansk (Danish)] #[http://pdc.wikipedia.org Deitsch (Pennsylvania German)] #[http://de.wikipedia.org Deutsch (German)] #[http://nv.wikipedia.org Diné bizaad (Navajo)] #[http://dv.wikipedia.org <span lang="dv" dir="rtl">ދިވެހި</span> (Divehi)] #[http://na.wikipedia.org dorerin Naoero (Nauruan)] #[http://dz.wikipedia.org <span lang="dz">ཇོང་ཁ</span> (Dzongkha)] #[http://lad.wikipedia.org Dzhudezmo (Ladino)] #[http://et.wikipedia.org Eesti (Estonian)] #[http://el.wikipedia.org <span lang="el">Ελληνικά</span> (Greek)] #[http://ang.wikipedia.org Englisc (Anglo-Saxon)] #[http://en.wikipedia.org English] #[http://es.wikipedia.org Español (Spanish)] #[http://eo.wikipedia.org Esperanto] #[http://eu.wikipedia.org Euskara (Basque)] #[http://to.wikipedia.org Faka Tonga (Tongan)] #[http://fa.wikipedia.org <span lang="fa" dir="rtl">‫فارسی‬</span> (Persian)] #[http://fo.wikipedia.org Føroyskt (Faroese)] #[http://fr.wikipedia.org Français (French)] #[http://frp.wikipedia.org Franco-provençal (Arpitan)] #[http://fy.wikipedia.org Frysk (West Frisian)] #[http://ff.wikipedia.org Fulfulde (Peul)] #[http://fur.wikipedia.org Furlan (Friulian)] #[http://ga.wikipedia.org Gaeilge (Irish)] #[http://gv.wikipedia.org Gaelg (Manx)] #[http://sm.wikipedia.org Gagana Samoa (Samoan)] #[http://gd.wikipedia.org Gàidhlig (Scots Gaelic)] #[http://gl.wikipedia.org Galego (Galician)] #[http://gu.wikipedia.org <span lang="gu">ગુજરાતી</span> (Gujarati)] #[http://got.wikipedia.org Gutisk (Gothic)] #[http://ko.wikipedia.org <span lang="ko">한국어</span> (Korean)] #[http://ha.wikipedia.org <span lang="ha" dir="rtl">هَوُسَ</span> (Hausa)] #[http://haw.wikipedia.org Hawai`i (Hawaiian)] #[http://hy.wikipedia.org <span lang="hy">Հայերեն</span> (Armenian)] #[http://hi.wikipedia.org <span lang="hi">हिन्दी</span> (Hindi)] #[http://ho.wikipedia.org Hiri Motu] #[http://hr.wikipedia.org Hrvatski (Croatian)] #[http://io.wikipedia.org Ido] #[http://ilo.wikipedia.org Ilokano] #[http://ig.wikipedia.org Igbo] #[http://ia.wikipedia.org Interlingua] #[http://ie.wikipedia.org Interlingue (ex Occidental)] #[http://iu.wikipedia.org <span lang="iu">ᐃᓄᒃᑎᑐᑦ</span> (Inuktitut)] #[http://ik.wikipedia.org Iñupiak (Inupiak)] #[http://os.wikipedia.org <span lang="os">Иронау</span> (Ossetian)] #[http://xh.wikipedia.org isiXhosa (Xhosa)] #[http://zu.wikipedia.org isiZulu (Zulu)] #[http://is.wikipedia.org Íslenska (Icelandic)] #[http://it.wikipedia.org Italiano (Italian)] #[http://he.wikipedia.org <span lang="he" dir="rtl">‫עברית‬</span> (Hebrew)] #[http://mh.wikipedia.org Kajin Majel (Marshallese)] #[http://kl.wikipedia.org Kalaallisut (Greenlandic)] #[http://xal.wikipedia.org <span lang="xal">Хальмг</span> (Kalmyk)] #[http://kn.wikipedia.org <span lang="kn">ಕನ್ನಡ</span> (Kannada)] #[http://kr.wikipedia.org Kanuri] #[http://pam.wikipedia.org Kapampangan] #[http://ka.wikipedia.org <span lang="ka">ქართული</span> (Georgian)] #[http://ks.wikipedia.org <span lang="ks">कश्मीरी</span>&nbsp;/ <span lang="ks-Arab" dir="rtl">كشميري</span> (Kashmiri)] #[http://csb.wikipedia.org Kaszëbsczi (Kashubian)] #[http://kk.wikipedia.org <span lang="kk">Қазақша</span> (Kazakh)] #[http://kw.wikipedia.org Kernewek&nbsp;/ Karnuack (Cornish)] #[http://kg.wikipedia.org Kikongo (Kongo)] #[http://ki.wikipedia.org Kikuyu] #[http://rw.wikipedia.org Kinyarwandi (Rwandi)] #[http://ky.wikipedia.org <span lang="ky-Latn">Kırgızca</span>&nbsp;/ <span lang="ky-Cyrl">Кыргызча</span> (Kyrgyz)] #[http://rn.wikipedia.org Kirundi] #[http://sw.wikipedia.org Kiswahili (Swahili)] #[http://km.wikipedia.org <span lang="km">ភាសាខ្មែរ</span> (Khmer)] #[http://kv.wikipedia.org <span lang="kv">Коми</span> (Komi)] #[http://ht.wikipedia.org Kreyol ayisyen (Haitian)] #[http://kj.wikipedia.org Kuanyama] #[http://ku.wikipedia.org <span lang="ku">Kurdî&nbsp;</span>/ <span lang="ku-Arab" dir="rtl">كوردی</span> (Kurdish)] #[http://ksh.wikipedia.org Ripoarisch (Ripuarian)] #[http://lo.wikipedia.org <span lang="lo">ພາສາລາວ</span> (Laotian)] #[http://la.wikipedia.org Latina (Latin)] #[http://lv.wikipedia.org Latviešu (Latvian)] #[http://lb.wikipedia.org Lëtzebuergesch (Luxembourgish)] #[http://lt.wikipedia.org Lietuvių (Lithuanian)] #[http://lij.wikipedia.org Líguru (Ligurian)] #[http://li.wikipedia.org Limburgs (Limburgish)] #[http://ln.wikipedia.org Lingála] #[http://jbo.wikipedia.org Lojban] #[http://lg.wikipedia.org Luganda] #[http://lmo.wikipedia.org Lumbaart (Lombard)] #[http://hu.wikipedia.org Magyar (Hungarian)] #[http://mk.wikipedia.org <span lang="mk">Македонски</span> (Macedonian)] #[http://mg.wikipedia.org Malagasy] #[http://ml.wikipedia.org <span lang="ml">മലയാളം</span> (Malayalam)] #[http://mi.wikipedia.org Māori] #[http://mr.wikipedia.org <span lang="mr">मराठी</span> (Marathi)] #[http://mo.wikipedia.org <span lang="mo">Молдовеняскэ</span> (Moldovan)] #[http://mn.wikipedia.org <span lang="mn">Монгол</span> (Mongolian)] #[http://mus.wikipedia.org Mvskoke (Creek)] #[http://my.wikipedia.org Myanmasa (Burmese)] #[http://nah.wikipedia.org Nāhuatl] #[http://fj.wikipedia.org Na Vosa Vakaviti (Fijian)] #[http://nl.wikipedia.org Nederlands (Dutch)] #[http://cr.wikipedia.org Nehiyaw (Cree)] #[http://ne.wikipedia.org <span lang="ne">नेपाली</span> (Nepali)] #[http://ja.wikipedia.org <span lang="ja">日本語</span> (Japanese)] #[http://nap.wikipedia.org Nnapulitano (Neapolitan)] #[http://ce.wikipedia.org <span lang="ce">Нохчийн</span> (Chechen)] #[http://nrm.wikipedia.org Nouormand&nbsp;/ Normaund (Norman)] #[http://pih.wikipedia.org Norfuk (Norfolk)] #[http://no.wikipedia.org Norsk bokmål (Norwegian Bokmål)] #[http://nn.wikipedia.org Norsk nynorsk (Norwegian Nynorsk)] #[http://oc.wikipedia.org Occitan] #[http://or.wikipedia.org <span lang="or">ଓଡ଼ିଆ</span> (Oriya)] #[http://om.wikipedia.org Oromifaa (Oromo)] #[http://ng.wikipedia.org Oshiwambo (Ndonga)] #[http://hz.wikipedia.org Otsiherero (Herero)] #[http://pa.wikipedia.org <span lang="pa">ਪਜਾਬੀ</span>&nbsp;/ <span lang="pa">पंजाबी</span>&nbsp;/ <span lang="pa-Arab" dir="rtl">پنجابي</span> (Punjabi)] #[http://pag.wikipedia.org Pangasinan (Pangasinan)] #[http://pi.wikipedia.org <span lang="pi-Latn">Pāli&nbsp;</span>/ <span lang="pi-Deva">पाऴि</span>] #[http://pap.wikipedia.org Papiamentu] #[http://ps.wikipedia.org <span lang="ps" dir="rtl">پښتو</span> (Pashto)] #[http://nds.wikipedia.org Plattdüütsch (Low Saxon)] #[http://pms.wikipedia.org Piemontèis (Piedmontese)] #[http://pl.wikipedia.org Polski (Polish)] #[http://pt.wikipedia.org Português (Portuguese)] #[http://ty.wikipedia.org Reo Mā`ohi (Tahitian)] #[http://ro.wikipedia.org Română (Romanian)] #[http://rm.wikipedia.org Rumantsch (Romansh)] #[http://rmy.wikipedia.org <span lang="rmy">Romani</span>&nbsp;/ <span lang="rmy-Deva">रोमानी</span>] #[http://qu.wikipedia.org Runa Simi (Quechua)] #[http://ru.wikipedia.org Русский (Russian)] #[http://se.wikipedia.org Sámegiella (Northern Sami)] #[http://sg.wikipedia.org Sängö] #[http://sa.wikipedia.org <span lang="sa">संस्कृत</span> (Sanskrit)] #[http://sc.wikipedia.org Sardu (Sardinian)] #[http://sco.wikipedia.org Scots] #[http://st.wikipedia.org seSotho (Southern Sotho)] #[http://tn.wikipedia.org Setswana (Tswana)] #[http://sq.wikipedia.org Shqip (Albanian)] #[http://ru-sib.wikipedia.org Siberian (Сибирской)] #[http://scn.wikipedia.org Sicilianu (Sicilian)] #[http://si.wikipedia.org <span lang="si">සිංහල</span> (Sinhalese)] #[http://simple.wikipedia.org Simple English] #[http://sd.wikipedia.org <span lang="sd">سنڌي</span> (Sindhi)] #[http://ceb.wikipedia.org Sinugboanong Binisaya (Cebuano)] #[http://ss.wikipedia.org SiSwati (Swati)] #[http://sk.wikipedia.org Slovenčina (Slovak)] #[http://sl.wikipedia.org Slovenščina (Slovenian)] #[http://so.wikipedia.org Soomaaliga (Somali)] #[http://sr.wikipedia.org Српски (Serbian)] #[http://sh.wikipedia.org <span lang="sh-Latn">Srpskohrvatski</span>&nbsp;/ <span lang="sh-Cyrl">Српскохрватски</span> (Serbo-Croatian)] #[http://fi.wikipedia.org Suomi (Finnish)] #[http://sv.wikipedia.org Svenska (Swedish)] #[http://ta.wikipedia.org <span lang="ta">தமிழ்</span> (Tamil)] #[http://tt.wikipedia.org Tatarça (Tatar)] #[http://te.wikipedia.org <span lang="te">తెలుగు</span> (Telugu)] #[http://tet.wikipedia.org Tetun (Tetum)] #[http://th.wikipedia.org <span lang="th">ไทย</span> (Thai)] #[http://vi.wikipedia.org Tiếng Việt (Vietnamese)] #[http://ti.wikipedia.org <span lang="ti">ትግርኛ</span> (Tigrinya)] #[http://tg.wikipedia.org <span lang="tg">Тоҷикӣ</span> (Tajik)] #[http://tlh.wikipedia.org tlhIngan-Hol (Klingon)] #[http://tpi.wikipedia.org Tok Pisin] #[http://chr.wikipedia.org <span lang="chr">ᏣᎳᎩ</span> (Cherokee)] #[http://chy.wikipedia.org Tsetsêhestâhese (Cheyenne)] #[http://tr.wikipedia.org Türkçe (Turkish)] #[http://tk.wikipedia.org <span lang="tk-Latn">Türkmençe&nbsp;/ </span><span lang="tk-Arab" dir="rtl">تركمن</span> (Turkmen)] #[http://tw.wikipedia.org Twi] #[http://udm.wikipedia.org <span lang="udm">Удмурт</span> (Udmurt)] #[http://uk.wikipedia.org <span lang="uk">Українська</span> (Ukrainian)] #[http://ur.wikipedia.org <span lang="ur" dir="rtl">اردو</span> (Urdu)] #[http://ug.wikipedia.org <span lang="ug-Arab" dir="rtl">ئۇيغۇرچە</span>&nbsp;/ <span lang="ug-Latn">Uyƣurqə</span> (Uyghur)] #[http://uz.wikipedia.org <span lang="uz-Cyrl">Ўзбек</span>&nbsp;/ <span lang="uz-Latn">Oʻzbekche</span> (Uzbek)] #[http://vec.wikipedia.org Vèneto (Venetian)] #[http://vo.wikipedia.org Volapük] #[http://fiu-vro.wikipedia.org Võro] #[http://wa.wikipedia.org Walon (Waloon)] #[http://vls.wikipedia.org West-Vlaoms (West Flemish)] #[http://war.wikipedia.org Winaray (Waray-Waray)] #[http://wo.wikipedia.org Wollof (Wolof)] #[http://ts.wikipedia.org Xitsonga (Tsonga)] #[http://ii.wikipedia.org <span lang="ii">ꆇꉙ</span> (Sichuan Yi)] #[http://yi.wikipedia.org <span lang="yi" dir="rtl">ייִדיש</span> (Yiddish)] #[http://yo.wikipedia.org Yorùbá] #[http://bat-smg.wikipedia.org Žemaitėška (Samogitian)] #[http://zh.wikipedia.org <span lang="zh">中文</span> (Chinese)]]] #[http://zh-yue.wikipedia.org <span lang="zh-tw">粵語</span> (Cantonese)] === Wiktionary === === Wikibooks === === Other === *[http://www.omegawiki.org/Main_Page Omegawiki] - multilingual resource in every language, with lexicological, terminological and thesaurus information. ==Active participants== Please show the laguages you have a working knowlege of. * [[User:CQ|CQ]] - [[Portal:English Language|English]] *... ''Join the [[Wikiversity:Interlingual Beta Club|Interlingual Beta Club]]'' == Other information == {{WikiversityUsers}} [[Category:Foreign Language Learning]] [[Category:Interlingual Beta Club]] [[Category:Languages]] [[Category: Multilingual Studies]] [[Category:Wikiversity portals]] [[el:Τμήμα:Μετάφραση και διερμηνεία]] 8ae2nlc38me34w9hhzo963ge4kcy0ap 2803504 2803472 2026-04-08T07:43:07Z CarlessParking 3064444 /* Languages */ 2803504 wikitext text/x-wiki [[Portal:Translation|Translation Department]] • <small>[[Portal:Multilingual Studies|Wikiversity Institute for Multilingual Studies]]</small> ==Subdivision news== * '''23 October 2006''' - Subdivision founded! * '''29 November 2006''' - Setting up at [http://beta.wikiversity.org/wiki/Translation beta.wikiversity] * ... == Learning Groups == See [[Portal:Multilingual Studies|Topic:Multilingual Studies]] for a more in-depth listing for Learning groups and topics within the larger [[Multilingual worksheet|Multilingulism Practicum]]. This is just a rough outline: [[School:Language and Literature]]: *[[Portal:Foreign Language Learning|Topic:Foreign Language Learning]] * ... [[School:Linguistics]]: *[[Topic:History of Linguistics|History of Linguistics]] *[[Portal:English Language|Topic:English Language]] *... [[School:Media studies|School:Media Studies]] *[[Wikiversity the Movie]] *[[Wikiversity the Movie/Multi-lingual|Wikiversity the Movie/multi-lingual]] *... [[School:Computer Science]] *[[Portal:Computational linguistics|Topic:Computational linguistics]] *[[Computer-assisted translation]] *... == Learning materials == [[Wikiversity:Translator's Handbook]]. * Introduction ''See [[Wikiversity:Essay Contest]]'' * [[Portal:Languages|Languages]] ''Will discuss [[:Category:Languages and Language families]] in detail covering'' * [[Topic:Linguistics]] * [[Portal:Computational linguistics|Topic:Lexicography]] * [[Portal:Translation|Appendix]] == Multilingualism at Wikiversity == [[Wikiversity:Multilingualism]] will be a centralized local resource for the English version of Wikiversity (en.wikiversity.org) ''about'' other versions of Wikiversity and beta projects. [[Wikiversity translations]]: *[[Wikiversity:Main Page|English]] *[[:de:Hauptseite]] [http://de.wikiversity.org Hauptseite] *[[:es:Portada]] [http://es.wikiversity.org Portada] Links to betas: *... == Wikiversity Translators == {{tl|translator}} @{{tl|translators}} ... {{translator}} <br><br> {{translators}} [[:Category:Wikiversity Translators]] {{construct}} == Wikimedia Service community == The following sections cover translation-related topics and list [[Wikiversity:Service|Translation services]] by and for the Wikimedia-Wikiversity [[Wikiversity:Service community]]. === Meta === [[m:Main Page|Meta]] has a cornucopia of resources for translators! [[m:Multilingualism]]: *[[m:Translators]] *[[m:Translation process]] *[[m:Interlingual coordination]] *[[m:Meta:Conventions for multilingualism]] *[[m:Wikipedia Machine Translation Project]] *... [[m:Category:Multilingualism]] *[[m:Category:Transbabel]] *[[m:Category:Translator's Templates]] *... <small>Note: We have adapted the meta Tranlator's Templates and are working on an adaptation of the [[Transbabel Multilingual Schema]] here at [[Wikiversity:Templates#Translation]] See [[Template:Translators]].</small> === Wikipedia === [[w:Main Page|Wikipedia]] ==== Articles ==== '''[[w:Meaning (linguistics)|Meanings]]:''' [[w:grammar|grammar]], [[w:semantics|semantics]], [[w:syntax|syntax]], [[w:idiom|idiom]], [[w:word|word]], [[w:phrase|phrase]], [[w:sentance|sentance]], [[w:source text|source text]], [[w:source language|source language]], [[w:target language|target language]], ... '''[[w:Language|Language]]:''' [[w:Natural language|Natural language]], [[w:Language family|Language family]], [[w:Linguistic typology|Linguistic typology]], [[w:Writing systems|Writing systems]], [[w:List of languages|List of languages]], [[w:List of official languages|List of official languages]], [[w:List of common phrases in various languages|List of common phrases in various languages]], ... '''[[w:Linguistics|Linguistics]]:''' [[w:Morphology (linguistics)|Morphology]], [[w:Etymology|Etymology]], [[w:Corpus linguistics|Corpus linguistics]], [[w:Text corpus|Text corpus]], [[w:Cognitive linguistics|Cognitive linguistics]], [[w:Computational linguistics|Computational linguistics]], [[w:Machine translation|Machine translation]], [[w:Speech recognition|Speech recognition]], [[w:Language recognition chart|Language recognition chart]], [[w:Part-of-speech tagging|Part-of-speech tagging]], ... '''[[w:Translation|Translation]]:''' [[w:translation dictionary|translation dictionary]], [[w:Translation process|Translation process]], [[w:Translation studies|Translation studies]], [[w:Translation unit|Translation unit]], [[w:Computer-assisted translation|Computer-assisted translation]], [[w:Machine translation|Machine translation]], ... ==== Projects ==== '''[[w:Wikipedia:Wikipedia Embassy|Wikipedia Embassy]]:''' *[[w:Wikipedia:Multilingual coordination|Wikipedia:Multilingual coordination]] *[[w:Wikipedia:Wikipedians]] *[[w:Wikipedia:Interwikimedia link]] *[[w:Wikipedia:Interlanguage links]] *... '''[[w:Wikipedia:WikiProject Linguistics|Wikipedia:WikiProject Linguistics]]:''' *[[w:Wikipedia:WikiProject Languages|WikiProject Languages]] *[[w:Wikipedia:WikiProject Language families|WikiProject Language families]] *[[w:Wikipedia:WikiProject Writing systems|WikiProject Writing systems]] *[[w:Wikipedia:WikiProject Latin alphabet|WikiProject Latin alphabet]] *[[w:Wikipedia:WikiProject Phonetics|WikiProject Phonetics]] *[[w:Wikipedia:WikiProject Constructed languages|WikiProject Constructed languages]] *[[w:Wikipedia:WikiProject Theoretical Linguistics|WikiProject Theoretical Linguistics]] '''[[w:Wikipedia:WikiProject Computer science|Wikipedia:WikiProject Computer science]]''' ==== Languages ==== List of Wikipedias in other languages: #[http://ab.wikipedia.org <span lang="ab-Cyrl">аҧсуа бысжѡа</span> (Abkhazian)] #[http://af.wikipedia.org Afrikaans] #[http://als.wikipedia.org Alemannisch (Alemannic)] #[http://am.wikipedia.org <span lang="am" dir="rtl">አማርኛ</span> (Amharic)] #[http://ar.wikipedia.org <span lang="ar" dir="rtl">‫العربية‬</span> (Arabic)] #[http://an.wikipedia.org Aragonés (Aragonese)] #[http://roa-rup.wikipedia.org Armâneashti (Aromanian)] #[http://as.wikipedia.org <span lang="as">অসমীয়া</span>&nbsp;/ <span lang="as-Latn">Asami</span> (Assamese)] #[http://arc.wikipedia.org <span lang="arc">ܣܘܪܬ</span> (Assyrian Neo-Aramaic)] #[http://ast.wikipedia.org Asturianu (Asturian)] #[http://gn.wikipedia.org Avañe'ẽ (Guarani)] #[http://av.wikipedia.org Авар (Avar)] #[http://ay.wikipedia.org Aymar (Aymara)] #[http://az.wikipedia.org <span lang="az">Azərbaycan dili</span>&nbsp;/ <span lang="az-Arab" dir="rtl">آذربايجان ديلی</span> (Azerbaijani)] #[http://id.wikipedia.org Bahasa Indonesia (Indonesian)] #[http://ms.wikipedia.org Bahasa Melayu (Malay)] #[http://bm.wikipedia.org Bamanankan (Bambara)] #[http://bn.wikipedia.org <span lang="bn">বাংলা</span> (Bengali)] #[http://zh-min-nan.wikipedia.org Bân-lâm-gú (Southern Min)] #[http://jv.wikipedia.org Basa Jawa (Javanese)] #[http://map-bms.wikipedia.org Basa Banyumasan (Banyumasan)] #[http://su.wikipedia.org Basa Sunda (Sundanese)] #[http://ba.wikipedia.org <span lang="ba">Башҡорт</span> (Bashkir)] #[http://be.wikipedia.org <span lang="be">Беларуская</span> (Belarusian)] #[http://bh.wikipedia.org <span lang="bh">भोजपुरी</span> (Bihari)] #[http://mt.wikipedia.org bil-Malti (Maltese)] #[http://bi.wikipedia.org Bislama] #[http://bo.wikipedia.org <span lang="bo">བོད་ཡིག</span>&nbsp;/ <span lang="bo-Latn">Bod skad</span> (Tibetan)] #[http://bs.wikipedia.org Bosanski (Bosnian)] #[http://br.wikipedia.org Brezhoneg (Breton)] #[http://bug.wikipedia.org Basa Ugi (Buginese)] #[http://bg.wikipedia.org <span lang="bg-Cyrl">Български</span> (Bulgarian)] #[http://ca.wikipedia.org Català (Catalan)] #[http://bcl.wikipedia.org Central Bikol (Bikol Central)] #[http://cbk-zam.wikipedia.org Chavacano (Chavacano de Zamboanga)] #[http://cho.wikipedia.org Cahta (Choctaw)] #[http://ch.wikipedia.org Chamorru (Chamorro)] #[http://cv.wikipedia.org <span lang="cv">Чӑваш</span> (Chuvash)] #[http://cs.wikipedia.org Čeština (Czech)] #[http://ny.wikipedia.org chiChewa] #[http://sn.wikipedia.org chiShona (Shona)] #[http://tum.wikipedia.org chiTumbuka (Tumbuka)] #[http://ve.wikipedia.org Chivenda (Venda)] #[http://co.wikipedia.org Corsu (Corsican)] #[http://za.wikipedia.org Cuengh (Zhuang)] #[http://cy.wikipedia.org Cymraeg (Welsh)] #[http://da.wikipedia.org Dansk (Danish)] #[http://pdc.wikipedia.org Deitsch (Pennsylvania German)] #[http://de.wikipedia.org Deutsch (German)] #[http://nv.wikipedia.org Diné bizaad (Navajo)] #[http://dv.wikipedia.org <span lang="dv" dir="rtl">ދިވެހި</span> (Divehi)] #[http://na.wikipedia.org dorerin Naoero (Nauruan)] #[http://dz.wikipedia.org <span lang="dz">ཇོང་ཁ</span> (Dzongkha)] #[http://lad.wikipedia.org Dzhudezmo (Ladino)] #[http://et.wikipedia.org Eesti (Estonian)] #[http://el.wikipedia.org <span lang="el">Ελληνικά</span> (Greek)] #[http://ang.wikipedia.org Englisc (Anglo-Saxon)] #[http://en.wikipedia.org English] #[http://es.wikipedia.org Español (Spanish)] #[http://eo.wikipedia.org Esperanto] #[http://eu.wikipedia.org Euskara (Basque)] #[http://to.wikipedia.org Faka Tonga (Tongan)] #[http://fa.wikipedia.org <span lang="fa" dir="rtl">‫فارسی‬</span> (Persian)] #[http://fo.wikipedia.org Føroyskt (Faroese)] #[http://fr.wikipedia.org Français (French)] #[http://frp.wikipedia.org Franco-provençal (Arpitan)] #[http://fy.wikipedia.org Frysk (West Frisian)] #[http://ff.wikipedia.org Fulfulde (Peul)] #[http://fur.wikipedia.org Furlan (Friulian)] #[http://ga.wikipedia.org Gaeilge (Irish)] #[http://gv.wikipedia.org Gaelg (Manx)] #[http://sm.wikipedia.org Gagana Samoa (Samoan)] #[http://gd.wikipedia.org Gàidhlig (Scots Gaelic)] #[http://gl.wikipedia.org Galego (Galician)] #[http://gu.wikipedia.org <span lang="gu">ગુજરાતી</span> (Gujarati)] #[http://got.wikipedia.org Gutisk (Gothic)] #[http://ko.wikipedia.org <span lang="ko">한국어</span> (Korean)] #[http://ha.wikipedia.org <span lang="ha" dir="rtl">هَوُسَ</span> (Hausa)] #[http://haw.wikipedia.org Hawai`i (Hawaiian)] #[http://hy.wikipedia.org <span lang="hy">Հայերեն</span> (Armenian)] #[http://hi.wikipedia.org <span lang="hi">हिन्दी</span> (Hindi)] #[http://ho.wikipedia.org Hiri Motu] #[http://hr.wikipedia.org Hrvatski (Croatian)] #[http://io.wikipedia.org Ido] #[http://ilo.wikipedia.org Ilokano] #[http://ig.wikipedia.org Igbo] #[http://ia.wikipedia.org Interlingua] #[http://ie.wikipedia.org Interlingue (ex Occidental)] #[http://iu.wikipedia.org <span lang="iu">ᐃᓄᒃᑎᑐᑦ</span> (Inuktitut)] #[http://ik.wikipedia.org Iñupiak (Inupiak)] #[http://os.wikipedia.org <span lang="os">Иронау</span> (Ossetian)] #[http://xh.wikipedia.org isiXhosa (Xhosa)] #[http://zu.wikipedia.org isiZulu (Zulu)] #[http://is.wikipedia.org Íslenska (Icelandic)] #[http://it.wikipedia.org Italiano (Italian)] #[http://he.wikipedia.org <span lang="he" dir="rtl">‫עברית‬</span> (Hebrew)] #[http://mh.wikipedia.org Kajin Majel (Marshallese)] #[http://kl.wikipedia.org Kalaallisut (Greenlandic)] #[http://xal.wikipedia.org <span lang="xal">Хальмг</span> (Kalmyk)] #[http://kn.wikipedia.org <span lang="kn">ಕನ್ನಡ</span> (Kannada)] #[http://kr.wikipedia.org Kanuri] #[http://pam.wikipedia.org Kapampangan] #[http://ka.wikipedia.org <span lang="ka">ქართული</span> (Georgian)] #[http://ks.wikipedia.org <span lang="ks">कश्मीरी</span>&nbsp;/ <span lang="ks-Arab" dir="rtl">كشميري</span> (Kashmiri)] #[http://csb.wikipedia.org Kaszëbsczi (Kashubian)] #[http://kk.wikipedia.org <span lang="kk">Қазақша</span> (Kazakh)] #[http://kw.wikipedia.org Kernewek&nbsp;/ Karnuack (Cornish)] #[http://kg.wikipedia.org Kikongo (Kongo)] #[http://ki.wikipedia.org Kikuyu] #[http://rw.wikipedia.org Kinyarwandi (Rwandi)] #[http://ky.wikipedia.org <span lang="ky-Latn">Kırgızca</span>&nbsp;/ <span lang="ky-Cyrl">Кыргызча</span> (Kyrgyz)] #[http://rn.wikipedia.org Kirundi] #[http://sw.wikipedia.org Kiswahili (Swahili)] #[http://km.wikipedia.org <span lang="km">ភាសាខ្មែរ</span> (Khmer)] #[http://kv.wikipedia.org <span lang="kv">Коми</span> (Komi)] #[http://ht.wikipedia.org Kreyol ayisyen (Haitian)] #[http://kj.wikipedia.org Kuanyama] #[http://ku.wikipedia.org <span lang="ku">Kurdî&nbsp;</span>/ <span lang="ku-Arab" dir="rtl">كوردی</span> (Kurdish)] #[http://ksh.wikipedia.org Ripoarisch (Ripuarian)] #[http://lo.wikipedia.org <span lang="lo">ພາສາລາວ</span> (Laotian)] #[http://la.wikipedia.org Latina (Latin)] #[http://lv.wikipedia.org Latviešu (Latvian)] #[http://lb.wikipedia.org Lëtzebuergesch (Luxembourgish)] #[http://lt.wikipedia.org Lietuvių (Lithuanian)] #[http://lij.wikipedia.org Líguru (Ligurian)] #[http://li.wikipedia.org Limburgs (Limburgish)] #[http://ln.wikipedia.org Lingála] #[http://jbo.wikipedia.org Lojban] #[http://lg.wikipedia.org Luganda] #[http://lmo.wikipedia.org Lumbaart (Lombard)] #[http://hu.wikipedia.org Magyar (Hungarian)] #[http://mk.wikipedia.org <span lang="mk">Македонски</span> (Macedonian)] #[http://mg.wikipedia.org Malagasy] #[http://ml.wikipedia.org <span lang="ml">മലയാളം</span> (Malayalam)] #[http://mi.wikipedia.org Māori] #[http://mr.wikipedia.org <span lang="mr">मराठी</span> (Marathi)] #[http://mo.wikipedia.org <span lang="mo">Молдовеняскэ</span> (Moldovan)] #[http://mn.wikipedia.org <span lang="mn">Монгол</span> (Mongolian)] #[http://mus.wikipedia.org Mvskoke (Creek)] #[http://my.wikipedia.org Myanmasa (Burmese)] #[http://nah.wikipedia.org Nāhuatl] #[http://fj.wikipedia.org Na Vosa Vakaviti (Fijian)] #[http://nl.wikipedia.org Nederlands (Dutch)] #[http://cr.wikipedia.org Nehiyaw (Cree)] #[http://ne.wikipedia.org <span lang="ne">नेपाली</span> (Nepali)] #[http://ja.wikipedia.org <span lang="ja">日本語</span> (Japanese)] #[http://nap.wikipedia.org Nnapulitano (Neapolitan)] #[http://ce.wikipedia.org <span lang="ce">Нохчийн</span> (Chechen)] #[http://nrm.wikipedia.org Nouormand&nbsp;/ Normaund (Norman)] #[http://pih.wikipedia.org Norfuk (Norfolk)] #[http://no.wikipedia.org Norsk bokmål (Norwegian Bokmål)] #[http://nn.wikipedia.org Norsk nynorsk (Norwegian Nynorsk)] #[http://oc.wikipedia.org Occitan] #[http://or.wikipedia.org <span lang="or">ଓଡ଼ିଆ</span> (Oriya)] #[http://om.wikipedia.org Oromifaa (Oromo)] #[http://ng.wikipedia.org Oshiwambo (Ndonga)] #[http://hz.wikipedia.org Otsiherero (Herero)] #[http://pa.wikipedia.org <span lang="pa">ਪਜਾਬੀ</span>&nbsp;/ <span lang="pa">पंजाबी</span>&nbsp;/ <span lang="pa-Arab" dir="rtl">پنجابي</span> (Punjabi)] #[http://pag.wikipedia.org Pangasinan (Pangasinan)] #[http://pi.wikipedia.org <span lang="pi-Latn">Pāli&nbsp;</span>/ <span lang="pi-Deva">पाऴि</span>] #[http://pap.wikipedia.org Papiamentu] #[http://ps.wikipedia.org <span lang="ps" dir="rtl">پښتو</span> (Pashto)] #[http://nds.wikipedia.org Plattdüütsch (Low Saxon)] #[http://pms.wikipedia.org Piemontèis (Piedmontese)] #[http://pl.wikipedia.org Polski (Polish)] #[http://pt.wikipedia.org Português (Portuguese)] #[http://ty.wikipedia.org Reo Mā`ohi (Tahitian)] #[http://ro.wikipedia.org Română (Romanian)] #[http://rm.wikipedia.org Rumantsch (Romansh)] #[http://rmy.wikipedia.org <span lang="rmy">Romani</span>&nbsp;/ <span lang="rmy-Deva">रोमानी</span>] #[http://qu.wikipedia.org Runa Simi (Quechua)] #[http://ru.wikipedia.org Русский (Russian)] #[http://se.wikipedia.org Sámegiella (Northern Sami)] #[http://sg.wikipedia.org Sängö] #[http://sa.wikipedia.org <span lang="sa">संस्कृत</span> (Sanskrit)] #[http://sc.wikipedia.org Sardu (Sardinian)] #[http://sco.wikipedia.org Scots] #[http://st.wikipedia.org seSotho (Southern Sotho)] #[http://tn.wikipedia.org Setswana (Tswana)] #[http://sq.wikipedia.org Shqip (Albanian)] #[http://ru-sib.wikipedia.org Siberian (Сибирской)] #[http://scn.wikipedia.org Sicilianu (Sicilian)] #[http://si.wikipedia.org <span lang="si">සිංහල</span> (Sinhalese)] #[http://simple.wikipedia.org Simple English] #[http://sd.wikipedia.org <span lang="sd">سنڌي</span> (Sindhi)] #[http://ceb.wikipedia.org Sinugboanong Binisaya (Cebuano)] #[http://ss.wikipedia.org SiSwati (Swati)] #[http://sk.wikipedia.org Slovenčina (Slovak)] #[http://sl.wikipedia.org Slovenščina (Slovenian)] #[http://so.wikipedia.org Soomaaliga (Somali)] #[http://sr.wikipedia.org Српски (Serbian)] #[http://sh.wikipedia.org <span lang="sh-Latn">Srpskohrvatski</span>&nbsp;/ <span lang="sh-Cyrl">Српскохрватски</span> (Serbo-Croatian)] #[http://fi.wikipedia.org Suomi (Finnish)] #[http://sv.wikipedia.org Svenska (Swedish)] #[http://tl.wikipia.org Tagalog] #[http://ta.wikipedia.org <span lang="ta">தமிழ்</span> (Tamil)] #[http://tt.wikipedia.org Tatarça (Tatar)] #[http://te.wikipedia.org <span lang="te">తెలుగు</span> (Telugu)] #[http://tet.wikipedia.org Tetun (Tetum)] #[http://th.wikipedia.org <span lang="th">ไทย</span> (Thai)] #[http://vi.wikipedia.org Tiếng Việt (Vietnamese)] #[http://ti.wikipedia.org <span lang="ti">ትግርኛ</span> (Tigrinya)] #[http://tg.wikipedia.org <span lang="tg">Тоҷикӣ</span> (Tajik)] #[http://tlh.wikipedia.org tlhIngan-Hol (Klingon)] #[http://tpi.wikipedia.org Tok Pisin] #[http://chr.wikipedia.org <span lang="chr">ᏣᎳᎩ</span> (Cherokee)] #[http://chy.wikipedia.org Tsetsêhestâhese (Cheyenne)] #[http://tr.wikipedia.org Türkçe (Turkish)] #[http://tk.wikipedia.org <span lang="tk-Latn">Türkmençe&nbsp;/ </span><span lang="tk-Arab" dir="rtl">تركمن</span> (Turkmen)] #[http://tw.wikipedia.org Twi] #[http://udm.wikipedia.org <span lang="udm">Удмурт</span> (Udmurt)] #[http://uk.wikipedia.org <span lang="uk">Українська</span> (Ukrainian)] #[http://ur.wikipedia.org <span lang="ur" dir="rtl">اردو</span> (Urdu)] #[http://ug.wikipedia.org <span lang="ug-Arab" dir="rtl">ئۇيغۇرچە</span>&nbsp;/ <span lang="ug-Latn">Uyƣurqə</span> (Uyghur)] #[http://uz.wikipedia.org <span lang="uz-Cyrl">Ўзбек</span>&nbsp;/ <span lang="uz-Latn">Oʻzbekche</span> (Uzbek)] #[http://vec.wikipedia.org Vèneto (Venetian)] #[http://vo.wikipedia.org Volapük] #[http://fiu-vro.wikipedia.org Võro] #[http://wa.wikipedia.org Walon (Waloon)] #[http://vls.wikipedia.org West-Vlaoms (West Flemish)] #[http://war.wikipedia.org Winaray (Waray-Waray)] #[http://wo.wikipedia.org Wollof (Wolof)] #[http://ts.wikipedia.org Xitsonga (Tsonga)] #[http://ii.wikipedia.org <span lang="ii">ꆇꉙ</span> (Sichuan Yi)] #[http://yi.wikipedia.org <span lang="yi" dir="rtl">ייִדיש</span> (Yiddish)] #[http://yo.wikipedia.org Yorùbá] #[http://bat-smg.wikipedia.org Žemaitėška (Samogitian)] #[http://zh.wikipedia.org <span lang="zh">中文</span> (Chinese)]]] #[http://zh-yue.wikipedia.org <span lang="zh-tw">粵語</span> (Cantonese)] === Wiktionary === === Wikibooks === === Other === *[http://www.omegawiki.org/Main_Page Omegawiki] - multilingual resource in every language, with lexicological, terminological and thesaurus information. ==Active participants== Please show the laguages you have a working knowlege of. * [[User:CQ|CQ]] - [[Portal:English Language|English]] *... ''Join the [[Wikiversity:Interlingual Beta Club|Interlingual Beta Club]]'' == Other information == {{WikiversityUsers}} [[Category:Foreign Language Learning]] [[Category:Interlingual Beta Club]] [[Category:Languages]] [[Category: Multilingual Studies]] [[Category:Wikiversity portals]] [[el:Τμήμα:Μετάφραση και διερμηνεία]] octkuyq9173226irxbn4nf5sk3ooy2h Portal:Southeast Asian languages 102 23729 2803474 2772171 2026-04-08T05:56:19Z CarlessParking 3064444 2803474 wikitext text/x-wiki <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Southeast Asian Languages Division</h2 > [[File:Map-World-Southeast-Asia.png|right|150px|Languages]] Welcome to the Southeast Asian Languages Division of the Schools of [[School:Language and Literature|Language and Literature]] and [[School:Linguistics|Linguistics]]. The primary aim of this division since its founding has been the study, preservation and promotion of the languages native to the Southeast Asian region through instruction, research and publication. </div> <div style="display:block;width:99%;float:left"> <div style="width:50%;display:block;float:left;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Courses and [[Portal:Learning Projects|Projects]]</h2 > [[Image:Crystal128-kanagram.svg|right|44px|]] * [[Southeast Asian Languages/Introduction| An introduction to the Southeast Asian languages]] :A brief description about the scope of the Division. * [[Southeast Asian Languages/Vietnamese_Language/Vietnamese One| Vietnamese 1]] :Vietnamese 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Philippine_Languages/Filipino One| Filipino 1]] :Filipino 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Bahasa Malaysia/Lesson:Introducing Yourself|Bahasa 1]] :Bahasa Malaysia 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Task List</h2 > [[Image:Nuvola apps korganizer.svg|right|44px|]] '''Philippine Languages Department''' * Curriculum formulation '''Bahasa Malaysia Department''' * Program organization '''Bahasa Indonesia Department''' * Recruitment of contributors '''Division-wide''' * Creation of more departments such as those for Vietnamese, Khmer, and Thai upon availability of writers and instructors. </div> </div> <div style="width:47%;display:block;float:right;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Departments</h2 > [[Image:Nuvola apps bookcase.svg|right|44px|]] The Division of Southeast Asian Languages is made up of several departments: * '''[[Southeast Asian Languages/Philippine Languages|The Philippine Languages Department]]''' * '''[[Southeast Asian Languages/Bahasa Malaysia|The Bahasa Malaysia Department]]''' * '''[[Southeast Asian Languages/Bahasa Indonesia|The Bahasa Indonesia Department]]''' * '''[[Southeast Asian Languages/Vietnamese Language|The Vietnamese Language Department]]''' </div > <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Division News</h2 > [[Image:Nuvola apps knewsticker.png|right|50px|]] * [[Topic:Southeast Asian Languages/News Archives|Archives]] * [[Topic:Southeast Asian Languages/Contributors and Students|Contributors and Students]] '''21 January 2007''' * The [[Southeast Asian Languages/Philippine Languages/Filipino One|Filipino 1]] class is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Malaysia|Bahasa Malaysia]] department is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Indonesia|Bahasa Indonesia]] department is currently in need of instructors and editors. '''22 January 2007''' * We are warmly inviting those with a working knowledge in Vietnamese, Bahasa Indonesia and Malaysia, Tagalog, Visayan, etc. language-wise or literary-wise to contribute ideas, lessons, and articles for this division ! '''15 February 2007''' * We are proud to launch the [[Southeast Asian Languages/Vietnamese Language|Vietnamese Language]] Department ! '''8 April 2026''' * We are proud to announce the launching of [[Southeast Asian Languages/Philippine Languages/Bikol|Bikol]] Department ! </div > </div > </div > __NOEDITSECTION__ __NOTOC__ [[Category:Southeast Asian languages| ]] cmq1ii4r13o168yfn48qt9ed79f3hj8 2803475 2803474 2026-04-08T05:56:54Z CarlessParking 3064444 2803475 wikitext text/x-wiki <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Southeast Asian Languages Division</h2 > [[File:Map-World-Southeast-Asia.png|right|150px|Languages]] Welcome to the Southeast Asian Languages Division of the Schools of [[School:Language and Literature|Language and Literature]] and [[School:Linguistics|Linguistics]]. The primary aim of this division since its founding has been the study, preservation and promotion of the languages native to the Southeast Asian region through instruction, research and publication. </div> <div style="display:block;width:99%;float:left"> <div style="width:50%;display:block;float:left;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Courses and [[Portal:Learning Projects|Projects]]</h2 > [[Image:Crystal128-kanagram.svg|right|44px|]] * [[Southeast Asian Languages/Introduction| An introduction to the Southeast Asian languages]] :A brief description about the scope of the Division. * [[Southeast Asian Languages/Vietnamese_Language/Vietnamese One| Vietnamese 1]] :Vietnamese 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Philippine_Languages/Filipino One| Filipino 1]] :Filipino 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Bahasa Malaysia/Lesson:Introducing Yourself|Bahasa 1]] :Bahasa Malaysia 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. </div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Task List</h2 > [[Image:Nuvola apps korganizer.svg|right|44px|]] '''Philippine Languages Department''' * Curriculum formulation '''Bahasa Malaysia Department''' * Program organization '''Bahasa Indonesia Department''' * Recruitment of contributors '''Division-wide''' * Creation of more departments such as those for Vietnamese, Khmer, and Thai upon availability of writers and instructors. </div> </div> <div style="width:47%;display:block;float:right;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Departments</h2 > [[Image:Nuvola apps bookcase.svg|right|44px|]] The Division of Southeast Asian Languages is made up of several departments: * '''[[Southeast Asian Languages/Philippine Languages|The Philippine Languages Department]]''' * '''[[Southeast Asian Languages/Bahasa Malaysia|The Bahasa Malaysia Department]]''' * '''[[Southeast Asian Languages/Bahasa Indonesia|The Bahasa Indonesia Department]]''' * '''[[Southeast Asian Languages/Vietnamese Language|The Vietnamese Language Department]]''' </div > <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Division News</h2 > [[Image:Nuvola apps knewsticker.png|right|50px|]] * [[Topic:Southeast Asian Languages/News Archives|Archives]] * [[Topic:Southeast Asian Languages/Contributors and Students|Contributors and Students]] '''21 January 2007''' * The [[Southeast Asian Languages/Philippine Languages/Filipino One|Filipino 1]] class is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Malaysia|Bahasa Malaysia]] department is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Indonesia|Bahasa Indonesia]] department is currently in need of instructors and editors. '''22 January 2007''' * We are warmly inviting those with a working knowledge in Vietnamese, Bahasa Indonesia and Malaysia, Tagalog, Visayan, etc. language-wise or literary-wise to contribute ideas, lessons, and articles for this division ! '''15 February 2007''' * We are proud to launch the [[Southeast Asian Languages/Vietnamese Language|Vietnamese Language]] Department ! '''8 April 2026''' * We are proud to announce the launching of [[Southeast Asian Languages/Philippine Languages/Bikol|Bikol Language]] Department ! </div > </div > </div > __NOEDITSECTION__ __NOTOC__ [[Category:Southeast Asian languages| ]] e0aaum80hksop0i7ep6th030j06wygg 2803505 2803475 2026-04-08T07:51:52Z CarlessParking 3064444 2803505 wikitext text/x-wiki <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Southeast Asian Languages Division</h2 > [[File:Map-World-Southeast-Asia.png|right|150px|Languages]] Welcome to the Southeast Asian Languages Division of the Schools of [[School:Language and Literature|Language and Literature]] and [[School:Linguistics|Linguistics]]. The primary aim of this division since its founding has been the study, preservation and promotion of the languages native to the Southeast Asian region through instruction, research and publication. </div> <div style="display:block;width:99%;float:left"> <div style="width:50%;display:block;float:left;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Courses and [[Portal:Learning Projects|Projects]]</h2 > [[Image:Crystal128-kanagram.svg|right|44px|]] * [[Southeast Asian Languages/Introduction| An introduction to the Southeast Asian languages]] :A brief description about the scope of the Division. * [[Southeast Asian Languages/Vietnamese_Language/Vietnamese One| Vietnamese 1]] :Vietnamese 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Philippine_Languages/Filipino One| Filipino 1]] :Filipino 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Bahasa Malaysia/Lesson:Introducing Yourself|Bahasa 1]] :Bahasa Malaysia 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. </div> * [[Southeast Asian Languages/Philippine_Languages/Bikol 1|Bikol 1]] :Filipino 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Task List</h2 > [[Image:Nuvola apps korganizer.svg|right|44px|]] '''Philippine Languages Department''' * Curriculum formulation '''Bahasa Malaysia Department''' * Program organization '''Bahasa Indonesia Department''' * Recruitment of contributors '''Division-wide''' * Creation of more departments such as those for Vietnamese, Khmer, and Thai upon availability of writers and instructors. </div> </div> <div style="width:47%;display:block;float:right;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Departments</h2 > [[Image:Nuvola apps bookcase.svg|right|44px|]] The Division of Southeast Asian Languages is made up of several departments: * '''[[Southeast Asian Languages/Philippine Languages|The Philippine Languages Department]]''' * '''[[Southeast Asian Languages/Bahasa Malaysia|The Bahasa Malaysia Department]]''' * '''[[Southeast Asian Languages/Bahasa Indonesia|The Bahasa Indonesia Department]]''' * '''[[Southeast Asian Languages/Vietnamese Language|The Vietnamese Language Department]]''' </div > <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Division News</h2 > [[Image:Nuvola apps knewsticker.png|right|50px|]] * [[Topic:Southeast Asian Languages/News Archives|Archives]] * [[Topic:Southeast Asian Languages/Contributors and Students|Contributors and Students]] '''21 January 2007''' * The [[Southeast Asian Languages/Philippine Languages/Filipino One|Filipino 1]] class is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Malaysia|Bahasa Malaysia]] department is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Indonesia|Bahasa Indonesia]] department is currently in need of instructors and editors. '''22 January 2007''' * We are warmly inviting those with a working knowledge in Vietnamese, Bahasa Indonesia and Malaysia, Tagalog, Visayan, etc. language-wise or literary-wise to contribute ideas, lessons, and articles for this division ! '''15 February 2007''' * We are proud to launch the [[Southeast Asian Languages/Vietnamese Language|Vietnamese Language]] Department ! '''8 April 2026''' * We are proud to announce the launching of [[Southeast Asian Languages/Philippine Languages/Bikol|Bikol Language]] Department ! </div > </div > </div > __NOEDITSECTION__ __NOTOC__ [[Category:Southeast Asian languages| ]] a4g5hcozm43ulu0gwzdj6g590x2a7at 2803506 2803505 2026-04-08T07:52:59Z CarlessParking 3064444 2803506 wikitext text/x-wiki <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Southeast Asian Languages Division</h2 > [[File:Map-World-Southeast-Asia.png|right|150px|Languages]] Welcome to the Southeast Asian Languages Division of the Schools of [[School:Language and Literature|Language and Literature]] and [[School:Linguistics|Linguistics]]. The primary aim of this division since its founding has been the study, preservation and promotion of the languages native to the Southeast Asian region through instruction, research and publication. </div> <div style="display:block;width:99%;float:left"> <div style="width:50%;display:block;float:left;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Courses and [[Portal:Learning Projects|Projects]]</h2 > [[Image:Crystal128-kanagram.svg|right|44px|]] * [[Southeast Asian Languages/Introduction| An introduction to the Southeast Asian languages]] :A brief description about the scope of the Division. * [[Southeast Asian Languages/Vietnamese_Language/Vietnamese One| Vietnamese 1]] :Vietnamese 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Philippine_Languages/Filipino One| Filipino 1]] :Filipino 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Bahasa Malaysia/Lesson:Introducing Yourself|Bahasa 1]] :Bahasa Malaysia 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language. * [[Southeast Asian Languages/Philippine_Languages/Bikol 1|Bikol 1]] :Bikol 1 is a course in which students are introduced to simple vocabulary, phrases, and elementary grammatical structures of the language.</div> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Task List</h2 > [[Image:Nuvola apps korganizer.svg|right|44px|]] '''Philippine Languages Department''' * Curriculum formulation '''Bahasa Malaysia Department''' * Program organization '''Bahasa Indonesia Department''' * Recruitment of contributors '''Division-wide''' * Creation of more departments such as those for Vietnamese, Khmer, and Thai upon availability of writers and instructors. </div> </div> <div style="width:47%;display:block;float:right;"> <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Departments</h2 > [[Image:Nuvola apps bookcase.svg|right|44px|]] The Division of Southeast Asian Languages is made up of several departments: * '''[[Southeast Asian Languages/Philippine Languages|The Philippine Languages Department]]''' * '''[[Southeast Asian Languages/Bahasa Malaysia|The Bahasa Malaysia Department]]''' * '''[[Southeast Asian Languages/Bahasa Indonesia|The Bahasa Indonesia Department]]''' * '''[[Southeast Asian Languages/Vietnamese Language|The Vietnamese Language Department]]''' </div > <div style="display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;"> <h2 style="padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;">Division News</h2 > [[Image:Nuvola apps knewsticker.png|right|50px|]] * [[Topic:Southeast Asian Languages/News Archives|Archives]] * [[Topic:Southeast Asian Languages/Contributors and Students|Contributors and Students]] '''21 January 2007''' * The [[Southeast Asian Languages/Philippine Languages/Filipino One|Filipino 1]] class is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Malaysia|Bahasa Malaysia]] department is currently in need of instructors and editors. * The [[Southeast Asian Languages/Bahasa Indonesia|Bahasa Indonesia]] department is currently in need of instructors and editors. '''22 January 2007''' * We are warmly inviting those with a working knowledge in Vietnamese, Bahasa Indonesia and Malaysia, Tagalog, Visayan, etc. language-wise or literary-wise to contribute ideas, lessons, and articles for this division ! '''15 February 2007''' * We are proud to launch the [[Southeast Asian Languages/Vietnamese Language|Vietnamese Language]] Department ! '''8 April 2026''' * We are proud to announce the launching of [[Southeast Asian Languages/Philippine Languages/Bikol|Bikol Language]] Department ! </div > </div > </div > __NOEDITSECTION__ __NOTOC__ [[Category:Southeast Asian languages| ]] ocbpsfcd9y7nfsq409gfdjlcalfltic Southeast Asian Languages/Introduction 0 23760 2803469 2742667 2026-04-08T04:07:23Z CarlessParking 3064444 2803469 wikitext text/x-wiki The languages of Southeast Asia consist of tongues belonging to more than one language family. Dominant families in the region include the Tai-Kadai (to which the Thai language belongs), Austronesian (such as Bahasa Malaysia and Indonesia as well as most Philippine languages), and Austro-Asiatic (e.g. Khmer and Vietnamese). With that said, cross-linguistic comparisons between these language families show a rather low affinity. In addition, the region is to some extent influenced by other language families due to cultural pressures, immigration, and historical colonization as well. The division aims to foster a more expanded and careful study of the languages indigenous to Southeast Asia, to aid foreigners to the countries within the region with useful words and phrases to get by, and finally to share with the world community the region's rich and charming literary traditions. Below is an incomplete list of the various languages used in Southeast Asia. Among these, the official languages are in bold: * Brunei: '''Malay''', indigenous Austronesian languages * Cambodia: '''Khmer''', Vietnamese, Chamic languages * Christmas Island: Malay * Cocos (Keeling) Islands: Cocos Malay * East Timor: '''Tetum''', Mambae, Makasae, Tukudede, Bunak, Galoli, Kemak, Fataluku, Baikeno, other Austronesian and Papuan languages * Indonesia: '''Indonesian''', Acehnese, Batak, Betawi, Sundanese, Javanese, Sasak, Tetum, "Dayak" languages, Minahasa, Toraja, Buginese, Halmahera, Ambonese, Ceramese, and many Papuan languages * Laos: '''Lao''', Hmong, Miao, Mien, Dao, Shan, and other Tibeto-Burman derived languages * Malaysia: '''Malay''', various indigenous languages (of the Orang Asli and indigenous peoples of Sabah and Sarawak) * Myanmar: '''Burmese''', Shan dialects, Karen dialects, Rakhine, Kachin, Chin, Mon, hilltribe languages * Philippines: '''Filipino''', '''English''', Tagalog, Cebuano, Hiligaynon, Waray-Waray, Ilokano, Kapampangan, Pangasinan, Bikol, Maranao, Maguindanao, Tausug, Kinaray-a, Chavacano (Spanish-based creole), other Philippine languages and dialects. * Singapore: '''Malay''', '''English''', '''Standard Chinese''', '''Tamil''', various Chinese languages * Thailand: '''Thai''', Isan, Shan, Lue, Phutai, Khmer, Mon, Mein, Hmong, Karen, Malay * Vietnam: '''Vietnamese''', Tay, Muong, Khmer, Nung, Hmong, Tai Dam, Malay, French creole [[Portal:Southeast Asian languages|Back to main page.]] [[Category:Introductions]] [[Category:Southeast Asian languages| ]] b4cgx8mk91ibnwnou0vkueaozbb23nu Complex Analysis 0 113381 2803526 2801642 2026-04-08T09:38:43Z Bert Niehaus 2387134 /* Chapter 5 - Holomorphic Functions */ 2803526 wikitext text/x-wiki == Introduction == {{mathematics}} [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** '''[[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Representation with Taylor Series|Representation with Taylor Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Representation%20with%20Taylor%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Representation%20with%20Taylor%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[/Examples for Power Series/]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|Cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Open mapping theorem|Open Mapping (and Connectedness) Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Open%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Open%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Holomorphic function/Criteria|Criteria for Holomorphy]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Criteria%20for%20Holomorphy&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Winding number|Winding number]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Winding%20number&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Winding%20number&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Residuals|Residuals]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/null-homologous|null-homologous]] * [[Complex Analysis/development in Laurent series|development in Laurent series]], * [[Complex Analysis/Isolated singularity|Isolated singularity]], * [[Complex Analysis/decomposition theorem|decomposition theorem]], * [[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]], *[[Riemann Removability Theorem]] * '''[[Complex Analysis/Residue Theorem|Residue Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Real integrals with residue theorem|Real integrals with residue theorem]] * '''[[Complex Analysis/Zeros and poles counting integral|Zeros and poles counting integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Zeros%20and%20poles%20counting%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Zeros%20and%20poles%20counting%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Rouché's theorem|Rouché's theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Rouch%C3%A9's%20theorem&author=Complex%20Analysis%20&language=en&audioslide=yes&shorttitle=Rouch%C3%A9's%20theorem&coursetitle=Complex%20Analysis%20 Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Meromorphic function|meromorphic function]] ===Riemann mapping theorem-automorphisms=== * '''[[Riemann mapping theorem|Riemann mapping theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Riemann%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Schwarz's Lemma|Schwarz's Lemma]] ** [[Complex Analysis/Automorphisms of the Unit Disk|<math>\mathrm{Aut}(\mathbb D)</math>]] *[[Complex Analysis/Harmonic_function|analytical and harmonic function ]] == Exercises == *Exercises for [[Complex Analysis/Exercises|Introduction to Complex Analysis ]] *[[Complex Analysis/Exercises/Sheet 1|Sheet 1]] *[[Complex Analysis/Exercises/Sheet 2|Sheet 2]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 3|Solution to Exercise 3]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 4|Solution to Exercise 4]] *[[Complex Analysis/Exercises/Sheet 3|Sheet 3]] *[[Complex Analysis/Exercises/Sheet 4|Sheet 4]] *[[Complex Analysis/Exercises/Sheet 5|Sheet 5]] *[[Complex Analysis/Exercises/Paper 1|Paper 1]] * [[Complex Analysis/Quiz]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> </noinclude> 0vdo8v6ja5soc1pxmff4osij2sy85b8 2803539 2803526 2026-04-08T10:31:19Z Bert Niehaus 2387134 /* Chapter 5 - Holomorphic Functions */ 2803539 wikitext text/x-wiki == Introduction == {{mathematics}} [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** '''[[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Representation with Taylor Series|Representation with Taylor Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Representation%20with%20Taylor%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Representation%20with%20Taylor%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Examples for Power Series/Approach|Approach for 1/z]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[/Examples for Power Series/]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|Cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Open mapping theorem|Open Mapping (and Connectedness) Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Open%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Open%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Holomorphic function/Criteria|Criteria for Holomorphy]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Criteria%20for%20Holomorphy&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Winding number|Winding number]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Winding%20number&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Winding%20number&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Residuals|Residuals]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/null-homologous|null-homologous]] * [[Complex Analysis/development in Laurent series|development in Laurent series]], * [[Complex Analysis/Isolated singularity|Isolated singularity]], * [[Complex Analysis/decomposition theorem|decomposition theorem]], * [[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]], *[[Riemann Removability Theorem]] * '''[[Complex Analysis/Residue Theorem|Residue Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Real integrals with residue theorem|Real integrals with residue theorem]] * '''[[Complex Analysis/Zeros and poles counting integral|Zeros and poles counting integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Zeros%20and%20poles%20counting%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Zeros%20and%20poles%20counting%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Rouché's theorem|Rouché's theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Rouch%C3%A9's%20theorem&author=Complex%20Analysis%20&language=en&audioslide=yes&shorttitle=Rouch%C3%A9's%20theorem&coursetitle=Complex%20Analysis%20 Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Meromorphic function|meromorphic function]] ===Riemann mapping theorem-automorphisms=== * '''[[Riemann mapping theorem|Riemann mapping theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Riemann%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Schwarz's Lemma|Schwarz's Lemma]] ** [[Complex Analysis/Automorphisms of the Unit Disk|<math>\mathrm{Aut}(\mathbb D)</math>]] *[[Complex Analysis/Harmonic_function|analytical and harmonic function ]] == Exercises == *Exercises for [[Complex Analysis/Exercises|Introduction to Complex Analysis ]] *[[Complex Analysis/Exercises/Sheet 1|Sheet 1]] *[[Complex Analysis/Exercises/Sheet 2|Sheet 2]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 3|Solution to Exercise 3]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 4|Solution to Exercise 4]] *[[Complex Analysis/Exercises/Sheet 3|Sheet 3]] *[[Complex Analysis/Exercises/Sheet 4|Sheet 4]] *[[Complex Analysis/Exercises/Sheet 5|Sheet 5]] *[[Complex Analysis/Exercises/Paper 1|Paper 1]] * [[Complex Analysis/Quiz]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> </noinclude> 86vjvmuchhyspujuaazjzelviuimo3i 2803541 2803539 2026-04-08T10:35:39Z Bert Niehaus 2387134 /* Singularity and Residues - Part 3 */ 2803541 wikitext text/x-wiki == Introduction == {{mathematics}} [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** '''[[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Representation with Taylor Series|Representation with Taylor Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Representation%20with%20Taylor%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Representation%20with%20Taylor%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Examples for Power Series/Approach|Approach for 1/z]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[/Examples for Power Series/]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|Cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Open mapping theorem|Open Mapping (and Connectedness) Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Open%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Open%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Holomorphic function/Criteria|Criteria for Holomorphy]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Criteria%20for%20Holomorphy&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Winding number|Winding number]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Winding%20number&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Winding%20number&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Residuals|Residue]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/null-homologous|null-homologous]] ** [[Complex Analysis/development in Laurent series|development in Laurent series]], ** [[Complex Analysis/Isolated singularity|Isolated singularity]], * '''[[Complex Analysis/decomposition theorem|Decomposition theorem]]''', * '''[[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]]''', *[[Riemann Removability Theorem]] * '''[[Complex Analysis/Residue Theorem|Residue Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Real integrals with residue theorem|Real integrals with residue theorem]] * '''[[Complex Analysis/Zeros and poles counting integral|Zeros and poles counting integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Zeros%20and%20poles%20counting%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Zeros%20and%20poles%20counting%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Rouché's theorem|Rouché's theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Rouch%C3%A9's%20theorem&author=Complex%20Analysis%20&language=en&audioslide=yes&shorttitle=Rouch%C3%A9's%20theorem&coursetitle=Complex%20Analysis%20 Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Meromorphic function|meromorphic function]] ===Riemann mapping theorem-automorphisms=== * '''[[Riemann mapping theorem|Riemann mapping theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Riemann%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Schwarz's Lemma|Schwarz's Lemma]] ** [[Complex Analysis/Automorphisms of the Unit Disk|<math>\mathrm{Aut}(\mathbb D)</math>]] *[[Complex Analysis/Harmonic_function|analytical and harmonic function ]] == Exercises == *Exercises for [[Complex Analysis/Exercises|Introduction to Complex Analysis ]] *[[Complex Analysis/Exercises/Sheet 1|Sheet 1]] *[[Complex Analysis/Exercises/Sheet 2|Sheet 2]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 3|Solution to Exercise 3]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 4|Solution to Exercise 4]] *[[Complex Analysis/Exercises/Sheet 3|Sheet 3]] *[[Complex Analysis/Exercises/Sheet 4|Sheet 4]] *[[Complex Analysis/Exercises/Sheet 5|Sheet 5]] *[[Complex Analysis/Exercises/Paper 1|Paper 1]] * [[Complex Analysis/Quiz]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> </noinclude> rfgd2gi6d38rk37lz751dn7i9jwog0i 2803545 2803541 2026-04-08T10:37:46Z Bert Niehaus 2387134 /* Singularity and Residues - Part 3 */ 2803545 wikitext text/x-wiki == Introduction == {{mathematics}} [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** '''[[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Representation with Taylor Series|Representation with Taylor Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Representation%20with%20Taylor%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Representation%20with%20Taylor%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Examples for Power Series/Approach|Approach for 1/z]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[/Examples for Power Series/]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|Cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Open mapping theorem|Open Mapping (and Connectedness) Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Open%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Open%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Holomorphic function/Criteria|Criteria for Holomorphy]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Criteria%20for%20Holomorphy&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Winding number|Winding number]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Winding%20number&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Winding%20number&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Residuals|Residue]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/null-homologous|null-homologous]] ** [[Complex Analysis/development in Laurent series|development in Laurent series]], ** [[Complex Analysis/Isolated singularity|Isolated singularity]], * '''[[Complex Analysis/Decomposition theorem|Decomposition theorem]]''', * '''[[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]]''', *[[Riemann Removability Theorem]] * '''[[Complex Analysis/Residue Theorem|Residue Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Real integrals with residue theorem|Real integrals with residue theorem]] * '''[[Complex Analysis/Zeros and poles counting integral|Zeros and poles counting integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Zeros%20and%20poles%20counting%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Zeros%20and%20poles%20counting%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Rouché's theorem|Rouché's theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Rouch%C3%A9's%20theorem&author=Complex%20Analysis%20&language=en&audioslide=yes&shorttitle=Rouch%C3%A9's%20theorem&coursetitle=Complex%20Analysis%20 Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Meromorphic function|meromorphic function]] ===Riemann mapping theorem-automorphisms=== * '''[[Riemann mapping theorem|Riemann mapping theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Riemann%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Schwarz's Lemma|Schwarz's Lemma]] ** [[Complex Analysis/Automorphisms of the Unit Disk|<math>\mathrm{Aut}(\mathbb D)</math>]] *[[Complex Analysis/Harmonic_function|analytical and harmonic function ]] == Exercises == *Exercises for [[Complex Analysis/Exercises|Introduction to Complex Analysis ]] *[[Complex Analysis/Exercises/Sheet 1|Sheet 1]] *[[Complex Analysis/Exercises/Sheet 2|Sheet 2]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 3|Solution to Exercise 3]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 4|Solution to Exercise 4]] *[[Complex Analysis/Exercises/Sheet 3|Sheet 3]] *[[Complex Analysis/Exercises/Sheet 4|Sheet 4]] *[[Complex Analysis/Exercises/Sheet 5|Sheet 5]] *[[Complex Analysis/Exercises/Paper 1|Paper 1]] * [[Complex Analysis/Quiz]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> </noinclude> dpec48ty3ucenz6t8ge4anu40ozg936 2803548 2803545 2026-04-08T10:39:18Z Bert Niehaus 2387134 /* Singularity and Residues - Part 3 */ 2803548 wikitext text/x-wiki == Introduction == {{mathematics}} [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** '''[[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Representation with Taylor Series|Representation with Taylor Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Representation%20with%20Taylor%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Representation%20with%20Taylor%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Examples for Power Series/Approach|Approach for 1/z]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[/Examples for Power Series/]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|Cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Open mapping theorem|Open Mapping (and Connectedness) Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Open%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Open%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Holomorphic function/Criteria|Criteria for Holomorphy]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Criteria%20for%20Holomorphy&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Winding number|Winding number]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Winding%20number&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Winding%20number&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Residue|Residue]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/null-homologous|null-homologous]] ** [[Complex Analysis/development in Laurent series|development in Laurent series]], ** [[Complex Analysis/Isolated singularity|Isolated singularity]], * '''[[Complex Analysis/Decomposition theorem|Decomposition theorem]]''', * '''[[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]]''', *[[Riemann Removability Theorem]] * '''[[Complex Analysis/Residue Theorem|Residue Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Real integrals with residue theorem|Real integrals with residue theorem]] * '''[[Complex Analysis/Zeros and poles counting integral|Zeros and poles counting integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Zeros%20and%20poles%20counting%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Zeros%20and%20poles%20counting%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Rouché's theorem|Rouché's theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Rouch%C3%A9's%20theorem&author=Complex%20Analysis%20&language=en&audioslide=yes&shorttitle=Rouch%C3%A9's%20theorem&coursetitle=Complex%20Analysis%20 Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Meromorphic function|meromorphic function]] ===Riemann mapping theorem-automorphisms=== * '''[[Riemann mapping theorem|Riemann mapping theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Riemann%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Schwarz's Lemma|Schwarz's Lemma]] ** [[Complex Analysis/Automorphisms of the Unit Disk|<math>\mathrm{Aut}(\mathbb D)</math>]] *[[Complex Analysis/Harmonic_function|analytical and harmonic function ]] == Exercises == *Exercises for [[Complex Analysis/Exercises|Introduction to Complex Analysis ]] *[[Complex Analysis/Exercises/Sheet 1|Sheet 1]] *[[Complex Analysis/Exercises/Sheet 2|Sheet 2]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 3|Solution to Exercise 3]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 4|Solution to Exercise 4]] *[[Complex Analysis/Exercises/Sheet 3|Sheet 3]] *[[Complex Analysis/Exercises/Sheet 4|Sheet 4]] *[[Complex Analysis/Exercises/Sheet 5|Sheet 5]] *[[Complex Analysis/Exercises/Paper 1|Paper 1]] * [[Complex Analysis/Quiz]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> </noinclude> 5enbqpbzo89ormujn04m8ml2s0asejf 2803551 2803548 2026-04-08T10:48:38Z Bert Niehaus 2387134 /* Singularity and Residues - Part 3 */ 2803551 wikitext text/x-wiki == Introduction == {{mathematics}} [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** '''[[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Representation with Taylor Series|Representation with Taylor Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Representation%20with%20Taylor%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Representation%20with%20Taylor%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Examples for Power Series/Approach|Approach for 1/z]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[/Examples for Power Series/]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|Cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Open mapping theorem|Open Mapping (and Connectedness) Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Open%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Open%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Holomorphic function/Criteria|Criteria for Holomorphy]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function/Criteria&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Criteria%20for%20Holomorphy&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Winding number|Winding number]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Winding%20number&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Winding%20number&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Residue|Residue]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/null-homologous|null-homologous]] ** [[Complex Analysis/development in Laurent series|development in Laurent series]], ** [[Complex Analysis/Isolated singularity|Isolated singularity]], * '''[[Complex Analysis/Decomposition theorem|Decomposition theorem]]''', * '''[[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]]''', * '''[[Riemann Removability Theorem|Riemann's theorem on removable singularities]]''' * '''[[Complex Analysis/Residue Theorem|Residue Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residue%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residue%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Real integrals with residue theorem|Real integrals with residue theorem]] * '''[[Complex Analysis/Zeros and poles counting integral|Zeros and poles counting integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Zeros%20and%20poles%20counting%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Zeros%20and%20poles%20counting%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Rouché's theorem|Rouché's theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Rouch%C3%A9's%20theorem&author=Complex%20Analysis%20&language=en&audioslide=yes&shorttitle=Rouch%C3%A9's%20theorem&coursetitle=Complex%20Analysis%20 Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Meromorphic function|meromorphic function]] ===Riemann mapping theorem-automorphisms=== * '''[[Riemann mapping theorem|Riemann mapping theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Riemann%20mapping%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20mapping%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/Schwarz's Lemma|Schwarz's Lemma]] ** [[Complex Analysis/Automorphisms of the Unit Disk|<math>\mathrm{Aut}(\mathbb D)</math>]] *[[Complex Analysis/Harmonic_function|analytical and harmonic function ]] == Exercises == *Exercises for [[Complex Analysis/Exercises|Introduction to Complex Analysis ]] *[[Complex Analysis/Exercises/Sheet 1|Sheet 1]] *[[Complex Analysis/Exercises/Sheet 2|Sheet 2]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 3|Solution to Exercise 3]] **[[Complex Analysis/Exercises/Sheet 2/Exercise 4|Solution to Exercise 4]] *[[Complex Analysis/Exercises/Sheet 3|Sheet 3]] *[[Complex Analysis/Exercises/Sheet 4|Sheet 4]] *[[Complex Analysis/Exercises/Sheet 5|Sheet 5]] *[[Complex Analysis/Exercises/Paper 1|Paper 1]] * [[Complex Analysis/Quiz]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> </noinclude> 369unkmt5i7win0xfbnd1oyo5gcyv82 Journal of Sport and Exercise Studies/Business, Politics and Sport 2011/The Effect of Animal Cruelty Allegations in Sport 0 123392 2803346 2723260 2026-04-07T15:07:24Z ~2026-21347-26 3064334 2803346 wikitext text/x-wiki [[User:Rhiannon Stephens|Rhiannon Stephens]], 2011<br> [[User:Rhiannon Stephens/The Effect of Animal Cruelty Allegations in Sport|Original copy]] [http://www.youtube.com/watch?v=80u2gVwmV3g&feature=youtu.be Presentation on Youtube] [[File:Bull-Riding-Szmurlo.jpg|thumb|400px|A bull riding contestant at the Calgary Stampede rodeo. Photo by Chuck Szmurlo 10/7/2007]] Animals have played a major role in sport, whether it has been for legal or illegal purposes. The organisations involved in utilising animals include rodeo, suck my balls and horse racing all have strict animal welfare policies in place to prevent animal cruelty allegations. However, controversy on this issue never far away and is brought about by animal rights organisations such as PETA, SAFE and SHARK. They claim that the techniques employed in these sports constitute animal cruelty. The sporting organisations claim that there is no information to support the claims made by animal cruelty organisations and that there information is based on propaganda. These are a major source of entertainment and they provide huge media and sponsorship deals. The negative side of this is that media coverage is the main stream of promotion for sports involved. Articles that are negative are more likely to influence the opinions of others than that of positive articles, which has the potential to damage the reputation, image and popularity of an organisation. In turn this has an effect on attendance, merchandising sponsorship and endorsement deals as organisations involved do not want to be seen as supporting animal cruelty. This article uses peer reviewed journals and organisational websites to describe the effects that animal cruelty controversies have on sporting organisations. There has been little research done in this area, more research needs to be done to see the full results of constant animal cruelty claims and how they affect sporting businesses. ==History of Animal Use in Sport== The use of animals in sport can be traced back to as far as when the ancient Greeks and Romans used horses in chariot races <ref name= "American">American Society for the Prevention of Cruelty to Animals, ‘Animal Cruelty’ in Learning to Give. 2008, viewed on 24 September 2011,<http://learningtogive.org/papers/paper359.html></ref>. Since then humans have continued to use animals in various sports. Rodeos have been present since the 18th century, not always considered a sport though <ref name="Graves">Graves, Melody, Ropes, Reins, And Rawhide: All About Rodeo, University of New Mexico Press, United States of America, 2006, p. 3-4</ref>. In the 1820’s and 1830’s Rodeos were considered informal events, the first Rodeo competition was held in 1872 in Cheyenne, Wyoming and Prescott, Arizona claimed having the first professional rodeo in which they charged admission and winning competitors were awarded <ref name="Graves"/>. Since then we have seen the emergence of the rodeos that are seen today. Organised horse racing dates back to the 17th century, and in the 20th century was one of the only sports to continue throughout both of the world wars and since then has emerged as an international sport<ref>Greenacres Stud, ‘History of Horse racing’, in Greenacres Stud. 2005, viewed on 25 September 2011, <http://www.greenacres-stud.com/horseracing.htm></ref>. Organisaed greyhound race meetings date back to 1926 in Britain, since then this event has become an extremely popular past time and has seen the emergence of a billion dollar industry<ref>Greyhound Board of Great Britain, ‘History of Greyhound Racing’, in Greyhound Board of Great Britain. 2009, viewed on 25 September 2011, <http://www.gbgb.org.uk/HistoryofGreyhoundRacing.aspx></ref> The billion dollar sporting industries that we see today are continuously modified to support legislation in order to prevent animal cruelty claims. The first animal cruelty laws were introduced in the 1870’s, and since then the laws have evolved into the legislation that is present today,<ref>MSPCA Angell, ‘MSPCA Law Enforcement’ on CRUELTY PREVENTION. 2006, viewed on 29 September 2011,<http://www.mspca.org/programs/cruelty-prevention/></ref>. After the release of such legislation, welfare controversies have surrounded these particular sports. Rodeos, horse and greyhoud racing are examples of sports that are legal to participate in as there are strict guidelines to ensure that the animals are protected. However, there are other sports that involve animals but to participate in blood sports such as cock fighting and dog fighting is illegal<ref>The Post and Courier, ‘Cock fighting illegal, but not gone’, in The Post and Courier. August 2008, viewed 18 October, <http://www.postandcourier.com/news/2008/aug/15/cockfighting_illegal_but_not_gone50928/></ref>. ==Opinions of the Media== Sports involving animals generate huge amounts of media coverage and sponsorship. Unfortunately, for the organisations involved media coverage is the main type of promotion for these sports and the organisations are not able to control the opinions that they share with the world <ref name="Bruce">Bruce, Toni & Tini, Tahlia ‘Unique crisis response strategies in sports public relations: Rugby league and the case for diversion’ Public Relations Review, Volume. 34, no. 2, 2008, pp 108-115.</ref>. Telstra's pull out was driven by Infrastructure Chief Ross Lambi who felt Telstra was over committed to sports. Negative publicity arises from animal cruelty claims that emerge from animal rights organisations such as People for Ethical Treatment of animals (PETA), SAFE and SHARK and they portray the sports in the worst possible way. The organisations believe that animals should not be used in sports as it constitutes animal cruelty <ref name = "SHARK">SHARK, ‘Forget the Myth’, in RodeoCruelty.com. November 2009, viewed on 3 September 2011, <http://www.sharkonline.org/?P=0000000349></ref>. The organisations also argue that the people involved in these sports do not follow the rules and regulations that are set out by the Government and the sporting bodies and therefore we should have total animal liberation<ref name= "PETA"> Wetz, Max, ‘Animal Defenders of Westchester’, in PETA and Rodeos. July 2008, viewed on 1 September 2001, <http://www.all-creatures.org/adow/art-rodeo-20030131.html></ref>. Rodeo itself attracts a significant amount of negative publicity as organisations believe that the events cause distress and injuries to the animals and they protest that rodeo’s should be banned<ref name="APRA"> Australian Professional Rodeo Association, ‘Animal Welfare- What does it mean?’ in Animal Welfare. n.d viewed on 1 September 2011, < http://prorodeo.asn.au/animals.htm></ref>. Animal rights activists try to discourage people from attending these events by convincing them that if they attend the event thay are supporting animal cruelty as. Statements such as this "Rodeo events are, no matter the gloss put on them, gratuitously violent acts against defenceless animals” are used to deter people from attending these events <ref name= "PETA">Wetz, Max, ‘Animal Defenders of Westchester’, in PETA and Rodeos. July 2008, viewed on 1 September 2001, <http://www.all-creatures.org/adow/art-rodeo-20030131.html><ref/>Issues are also raised in horse racing as animal rights organisations believe that hitting the horse with a whip constitutes animal cruelty. other concerns arise as they believe that horses are raced to young and that they are not mature enough and this causes trouble to their skeletal system. It is also believed that the horses suffer physically and mentally from being locked in a stable and the food that they consume causes them stomach ulcers<ref>Animals Australia, ‘the glitz the glam our... the grim reality’ in Horse Racing. August 2010, viewed 5 September 2011, < http://www.animalsaustralia.org/issues/horse_racing.php#toc4></ref>. The greyhound racing industry faces many allegations of animal cruelty or abuse as animal cruelty organisations believe that the greyhounds are suffering as they are do not race willingly and that if they are not winners they are disposed of<ref>Animal Aid, ‘Greyhound Racing’, in Animals in Sport.2011 viewed on 17 October 2011, <http://www.animalaid.org.uk/h/n/YOUTH/sport//1877//></ref>. By using these examples it can be seen that theses organisations are plagued by animal welfare controversy. {{center top}} {| border=1 cellspacing=5 cellpadding=5 | '''Media Portrayal''' | '''Sporting Organisation Portrayal''' |- | [[File:Calf_Roping_Rodeo_3.JPG‎|thumb|200px| 2005 Calf Roping. Photo by Ethelred]] This image is used by animal rights organisations to discourage people from going to watch these events. Saying that this is how the event unfolds every time. | [[File:Cowgirl Lasso 4889218394.jpg|thumb|200px|Woman roping a calf at the Buffalo Bill Cody Stampede Rodeo. Photo by C.G.P Grey]] This image would be used by organisations and is more appropriate to use as it does not imply that either animal is under any stress. |} {{center bottom}} == The Effect of Animal Cruelty Allegations== There has been and still is a significant amount of negative media surrounding sporting organisations such as the Australian Professional Rodeo Association (APRA), the Australian Racing Board (ARB) and Greyhound Racing South Australia (GRSA)over animal cruelty claims. This can cause problems for organisations as they spend more time reacting to unplanned events or accusations that negatively affect the attitudes of the public, rather than being proactive and trying to create positive attitudes within communities <ref name= "Bruce"/>. These statements could have serious implications for businesses involved in these sports such as rodeo, horse racing and greyhound racing. As the Result of people making statements about animal cruelty have the potential to have a negative impact on organisations as it has been proven that people have a tendency to remember more facts from negative articles than that of positive articles<ref name= "Funk">Funk, Daniel C & Pritchard, Mark P ‘Sport Publicity: Commitment’s moderation of message effects’ Journal of Business Research, vol 59, 2006 pp 613-621</ref>. The allegations made by other organisations are similar to that of PETA and SAFE. The result of the constant negative media can seriously damage the reputation, image and popularity of an organisation<ref name= "Bruce"/>. This then has an effect on areas that involve attendance, merchandising, sponsorship and endorsement deals. When expressing their opinions on animal cruelty organisations are discouraging people from attending events which could effect the ability for sporting businesses to make a profit<ref name= "Bruce"/>. An example of when sponsorship was removed is when Telstra removed sponsorship from a local rodeo in Corryong due to allegations of animal cruelty<ref>ABC News, ‘Telstra bucks sponsorship over rodeo cruelty fears’, in ABC News, September 2007, viewed on 4 September 2011, < http://www.abc.net.au/news/2007-09-19/telstra-bucks-sponsorship-over-rodeo-cruelty-fears/674208></ref>. . Another instance when sponsorship was removes was when Choice Hotels dropped their sponsorship of the National High School Rodeo Association (NHRSA) because an animal rights group alleging that by sponsoring the event they were supporting animal cruelty <ref name = "SHARK"/>.The Loss of sponsorship has an affect on sporting businesses as they are losing funds that are essential to hold and run events. Therefore the way that the organisation is portrayed in the media can have major impacts on sporting businesses. Sponsorship of an event is supposed to have a positive effect on the corporate image of an organisation <ref name= "Wilson">Wilson, Bradley, Stavros, Constantino & Westburg, Kate ‘A sport crisis typology: establishing a pathway for future research’, International Journal of Sport Management & marketing, vol. 7, no. 1, 2010, p 21</ref>. The association of a brand with sport can have negative effect if the sporting organisation generates a lot of negative media attention. This has the potential cause trouble for the sports that are involved in competitions which involve animal cruelty claims. Sponsorship of these events could be viewed as risk taking behaviour <ref name="Wilson"/>. Sponsorship is a fundamental ingredient for a successful event. It is a major stream of revenue for the organisers of an event and the withdrawal of sponsorship can have major impacts on an event and cause financial difficulties for organisations<ref>Westburg, Kate, Stavros, Kate & Wilson, Bradley ‘The impact of degenerative episodes on the sponsorship B2B relationship: Implications for brand management’ Journal of Industrial Marketing Management, vol 40, 2011, pp 603-611</ref>. There was controversy amongst the Greyhound racing industry when apparently not one representative from the McGrath Foundation turned up to receive a check from the greyhound racing industry. This was due to the public perceptions that by being associated with this organisation they support animal cruelty. However, they still wanted to receive the money, they just did not want to do it in the public eye <ref>Animals Australia, ‘The Australian Greyhound Racing Industry: The McGrath Foundation’, in Unleashed. March 2011, viewed 4 September 2011, <http://www.unleashed.org.au/community/forum/topic.php?t=4724></ref>. This affects the business of greyhound racing as people do not want to be involved or associated with sports that attract negative publicity as they do not want to be seen as supporting animal cruelty. Many other concerns arise as animal rights activists believe that hitting the horse with a whip is considered to be animal cruelty<ref name="ARB"/>. These allegations could possibly have an effect on the commodification of this sport as the activists discourage people from attending these events or placing bets <ref>Animals Australia, ‘the glitz the glam our... the grim reality’ in Horse Racing. August 2010, viewed 5 September 2011, < http://www.animalsaustralia.org/issues/horse_racing.php#toc4></ref>. They convince people that if they attend an event they are supporting animal cruelty, this is the same as in Rodeo. The affect of the constant scrutiny resulted in the ARB changing the rules of racing as these issues were raised at all levels of competition. Jockeys are no longer aloud to raise whips above their shoulder, also whips must be padded <ref name= "ARB"> Australian Racing Board, ‘Whip Reform Stays’, in Australian Racing Board. September 2009, viewed on 1 September 2011, <http://www.australianracingboard.com.au/Media.whip-reform-stays></ref>. Within 5 weeks of changing the rules the ARB believed that there had been a substantial improvement in the attitudes and practices of this sport <ref name="ARB"/>. This has had a positive impact on the horse racing industry. However, the opinions of the media still effect areas such as sponsorship for sporting organisations. {{center top}} {| border=1 cellspacing=5 cellpadding=5 |- | [[File:Horse-racing-1.jpg|thumb|300px|center|Horse racing at Galopp Riem 05/06/2005, Munich, Germany.Photo by Softeis.]] | [[File:Greyhound Racing 2 amk.jpg|thumb|350px|center|English: Greyhound racing 18/05/2008.Phot by AngMokio]] |} {{center bottom}} ==How Organisations Respond to Such Allegations == In defence to the animal cruelty claims the APRA argue that there is no information to support the claims that animals are treated in an inhumane way. They also believe that the statements that are made by groups such as PETA and SAFE are fabricated to support their personal emotions regarding the sport<ref name="APRA"/>. In repsonsding to animal cruelty claims in rodeo the APRA said that "The sport has had to become more transparent and proactive if it was to survive under today’s expectations"<ref name= "Stephens">Stephens, Rhiannon, 'Email interview transcript with Australian Professional Rodeo Association and Greyhound Racing Australia' in User_talk:Rhiannon Stephens. 2011, <http://en.wikiversity.org/wiki/User_talk:Rhiannon_Stephens/The_Effect_of_Animal_Cruelty_Allegations_in_Sport></ref> Wikiversity user:Rhiannon Stephens talk page</ref>. Statistics of injured livestock are taken at events and forwarded on to animal welfare regulatory bodies so that they can compare them with their own records. The APRA have also stated on their website that they value the animals that they use and follow strict animal welfare policies <ref name="APRA"/>. This website also explains for people that do not understand the sport of Rodeo that the use of these animals in such a way does not involve animal cruelty<ref>Australian Professional Rodeo Association, ‘The Facts Concerning the Care and Treatment of Professional Rodeo Livestock’, in Animal Welfare .n.d, viewed on 5 September 2011, <http://prorodeo.asn.au/forms/AnimalWelfare.pdf></ref>. As seen in the horse racing example the ARB responded to animal cruelty allegations by changing the rules and regulations regarding whips. They believe that by having done this they have seen an improvement in the attitudes of people around the sport, therefore this has had a positive impact. In order to prevent allegations GRSA have partnered with Technical and Further Education (TAFE) and the University of South Australia to ensure that their animals receive the best possible care(Email). Also the dogs are placed in foster homes so that they learn how to become pets and once they are finished racing are placed into permanent homes<ref name= "Stephens"/>. They also ensure people that they put animal welfare at the heart of what they do<ref name ="Stephens"/>. All organisations that are involved in these sports have made sure that policies in place to protect the animals that are involved. ==Conclusion== Animals have been used in sports since the ancient Greek and Roman chariot races, and continue to be used in professional sports today<ref name= "American"/>. The emergence of animals in sport brought about animal welfare legislation in the 1870’s, which provides the laws for which animals are to be treated<ref name= "Graves"/>. These sports are a major source of entertainment and provide huge media and sponsorship deals. The organisations are not able to control the opinions that are expressed in the media <ref name= "Bruce"/>. The negative publicity that is generated has huge impacts on sporting organisations<ref name= "Bruce"/>. In the media, animal rights organisations claim that the use of animals in sport is what constitutes animal cruelty and we should have total animal liberation. In stating their opinions the animal rights organisations discourage people from attending events and betting. These articles are negative and people are more likely to remember the facts from these than positive articles<ref name="Funk"/>. The negative media coverage damages the reputation, image and popularity of an organisation which in turn affects attendance, merchandising, sponsorship and endorsement deals as other people and organisations do not want to be seen as supporting animal cruelty<ref name= "Bruce"/>. This is a major concern where sponsorship is concerned as association with an organisation can have a negative effect if it attracts substantial amounts of negative publicity, in which these sports do<ref name="Bruce"/>. The statements that are made in the media to the public have a huge impact on the businesses that are involved and this then affects all aspects of the sport<ref name= "Bruce"/>. The organisations involved claim that there is no information to support the claims made by animal rights organisations that their opinions are fabricated to represent their personal emotions regarding the sport. ==References== {{reflist}} {{CourseCat}} 0qc8gt8viu1m8enc0tpbhyjw9kxkyi8 2803470 2803346 2026-04-08T04:15:59Z PieWriter 3039865 Reverted edits by [[Special:Contributions/~2026-21347-26|~2026-21347-26]] ([[User_talk:~2026-21347-26|talk]]) to last version by [[User:2405:6E00:23E:1E7C:9802:2F73:B4B2:D28D|2405:6E00:23E:1E7C:9802:2F73:B4B2:D28D]] using [[Wikiversity:Rollback|rollback]] 2723260 wikitext text/x-wiki [[User:Rhiannon Stephens|Rhiannon Stephens]], 2011<br> [[User:Rhiannon Stephens/The Effect of Animal Cruelty Allegations in Sport|Original copy]] [http://www.youtube.com/watch?v=80u2gVwmV3g&feature=youtu.be Presentation on Youtube] [[File:Bull-Riding-Szmurlo.jpg|thumb|400px|A bull riding contestant at the Calgary Stampede rodeo. Photo by Chuck Szmurlo 10/7/2007]] Animals have played a major role in sport, whether it has been for legal or illegal purposes. The organisations involved in utilising animals include rodeo, greyhound and horse racing all have strict animal welfare policies in place to prevent animal cruelty allegations. However, controversy on this issue never far away and is brought about by animal rights organisations such as PETA, SAFE and SHARK. They claim that the techniques employed in these sports constitute animal cruelty. The sporting organisations claim that there is no information to support the claims made by animal cruelty organisations and that there information is based on propaganda. These are a major source of entertainment and they provide huge media and sponsorship deals. The negative side of this is that media coverage is the main stream of promotion for sports involved. Articles that are negative are more likely to influence the opinions of others than that of positive articles, which has the potential to damage the reputation, image and popularity of an organisation. In turn this has an effect on attendance, merchandising sponsorship and endorsement deals as organisations involved do not want to be seen as supporting animal cruelty. This article uses peer reviewed journals and organisational websites to describe the effects that animal cruelty controversies have on sporting organisations. There has been little research done in this area, more research needs to be done to see the full results of constant animal cruelty claims and how they affect sporting businesses. ==History of Animal Use in Sport== The use of animals in sport can be traced back to as far as when the ancient Greeks and Romans used horses in chariot races <ref name= "American">American Society for the Prevention of Cruelty to Animals, ‘Animal Cruelty’ in Learning to Give. 2008, viewed on 24 September 2011,<http://learningtogive.org/papers/paper359.html></ref>. Since then humans have continued to use animals in various sports. Rodeos have been present since the 18th century, not always considered a sport though <ref name="Graves">Graves, Melody, Ropes, Reins, And Rawhide: All About Rodeo, University of New Mexico Press, United States of America, 2006, p. 3-4</ref>. In the 1820’s and 1830’s Rodeos were considered informal events, the first Rodeo competition was held in 1872 in Cheyenne, Wyoming and Prescott, Arizona claimed having the first professional rodeo in which they charged admission and winning competitors were awarded <ref name="Graves"/>. Since then we have seen the emergence of the rodeos that are seen today. Organised horse racing dates back to the 17th century, and in the 20th century was one of the only sports to continue throughout both of the world wars and since then has emerged as an international sport<ref>Greenacres Stud, ‘History of Horse racing’, in Greenacres Stud. 2005, viewed on 25 September 2011, <http://www.greenacres-stud.com/horseracing.htm></ref>. Organisaed greyhound race meetings date back to 1926 in Britain, since then this event has become an extremely popular past time and has seen the emergence of a billion dollar industry<ref>Greyhound Board of Great Britain, ‘History of Greyhound Racing’, in Greyhound Board of Great Britain. 2009, viewed on 25 September 2011, <http://www.gbgb.org.uk/HistoryofGreyhoundRacing.aspx></ref> The billion dollar sporting industries that we see today are continuously modified to support legislation in order to prevent animal cruelty claims. The first animal cruelty laws were introduced in the 1870’s, and since then the laws have evolved into the legislation that is present today,<ref>MSPCA Angell, ‘MSPCA Law Enforcement’ on CRUELTY PREVENTION. 2006, viewed on 29 September 2011,<http://www.mspca.org/programs/cruelty-prevention/></ref>. After the release of such legislation, welfare controversies have surrounded these particular sports. Rodeos, horse and greyhoud racing are examples of sports that are legal to participate in as there are strict guidelines to ensure that the animals are protected. However, there are other sports that involve animals but to participate in blood sports such as cock fighting and dog fighting is illegal<ref>The Post and Courier, ‘Cock fighting illegal, but not gone’, in The Post and Courier. August 2008, viewed 18 October, <http://www.postandcourier.com/news/2008/aug/15/cockfighting_illegal_but_not_gone50928/></ref>. ==Opinions of the Media== Sports involving animals generate huge amounts of media coverage and sponsorship. Unfortunately, for the organisations involved media coverage is the main type of promotion for these sports and the organisations are not able to control the opinions that they share with the world <ref name="Bruce">Bruce, Toni & Tini, Tahlia ‘Unique crisis response strategies in sports public relations: Rugby league and the case for diversion’ Public Relations Review, Volume. 34, no. 2, 2008, pp 108-115.</ref>. Telstra's pull out was driven by Infrastructure Chief Ross Lambi who felt Telstra was over committed to sports. Negative publicity arises from animal cruelty claims that emerge from animal rights organisations such as People for Ethical Treatment of animals (PETA), SAFE and SHARK and they portray the sports in the worst possible way. The organisations believe that animals should not be used in sports as it constitutes animal cruelty <ref name = "SHARK">SHARK, ‘Forget the Myth’, in RodeoCruelty.com. November 2009, viewed on 3 September 2011, <http://www.sharkonline.org/?P=0000000349></ref>. The organisations also argue that the people involved in these sports do not follow the rules and regulations that are set out by the Government and the sporting bodies and therefore we should have total animal liberation<ref name= "PETA"> Wetz, Max, ‘Animal Defenders of Westchester’, in PETA and Rodeos. July 2008, viewed on 1 September 2001, <http://www.all-creatures.org/adow/art-rodeo-20030131.html></ref>. Rodeo itself attracts a significant amount of negative publicity as organisations believe that the events cause distress and injuries to the animals and they protest that rodeo’s should be banned<ref name="APRA"> Australian Professional Rodeo Association, ‘Animal Welfare- What does it mean?’ in Animal Welfare. n.d viewed on 1 September 2011, < http://prorodeo.asn.au/animals.htm></ref>. Animal rights activists try to discourage people from attending these events by convincing them that if they attend the event thay are supporting animal cruelty as. Statements such as this "Rodeo events are, no matter the gloss put on them, gratuitously violent acts against defenceless animals” are used to deter people from attending these events <ref name= "PETA">Wetz, Max, ‘Animal Defenders of Westchester’, in PETA and Rodeos. July 2008, viewed on 1 September 2001, <http://www.all-creatures.org/adow/art-rodeo-20030131.html><ref/>Issues are also raised in horse racing as animal rights organisations believe that hitting the horse with a whip constitutes animal cruelty. other concerns arise as they believe that horses are raced to young and that they are not mature enough and this causes trouble to their skeletal system. It is also believed that the horses suffer physically and mentally from being locked in a stable and the food that they consume causes them stomach ulcers<ref>Animals Australia, ‘the glitz the glam our... the grim reality’ in Horse Racing. August 2010, viewed 5 September 2011, < http://www.animalsaustralia.org/issues/horse_racing.php#toc4></ref>. The greyhound racing industry faces many allegations of animal cruelty or abuse as animal cruelty organisations believe that the greyhounds are suffering as they are do not race willingly and that if they are not winners they are disposed of<ref>Animal Aid, ‘Greyhound Racing’, in Animals in Sport.2011 viewed on 17 October 2011, <http://www.animalaid.org.uk/h/n/YOUTH/sport//1877//></ref>. By using these examples it can be seen that theses organisations are plagued by animal welfare controversy. {{center top}} {| border=1 cellspacing=5 cellpadding=5 | '''Media Portrayal''' | '''Sporting Organisation Portrayal''' |- | [[File:Calf_Roping_Rodeo_3.JPG‎|thumb|200px| 2005 Calf Roping. Photo by Ethelred]] This image is used by animal rights organisations to discourage people from going to watch these events. Saying that this is how the event unfolds every time. | [[File:Cowgirl Lasso 4889218394.jpg|thumb|200px|Woman roping a calf at the Buffalo Bill Cody Stampede Rodeo. Photo by C.G.P Grey]] This image would be used by organisations and is more appropriate to use as it does not imply that either animal is under any stress. |} {{center bottom}} == The Effect of Animal Cruelty Allegations== There has been and still is a significant amount of negative media surrounding sporting organisations such as the Australian Professional Rodeo Association (APRA), the Australian Racing Board (ARB) and Greyhound Racing South Australia (GRSA)over animal cruelty claims. This can cause problems for organisations as they spend more time reacting to unplanned events or accusations that negatively affect the attitudes of the public, rather than being proactive and trying to create positive attitudes within communities <ref name= "Bruce"/>. These statements could have serious implications for businesses involved in these sports such as rodeo, horse racing and greyhound racing. As the Result of people making statements about animal cruelty have the potential to have a negative impact on organisations as it has been proven that people have a tendency to remember more facts from negative articles than that of positive articles<ref name= "Funk">Funk, Daniel C & Pritchard, Mark P ‘Sport Publicity: Commitment’s moderation of message effects’ Journal of Business Research, vol 59, 2006 pp 613-621</ref>. The allegations made by other organisations are similar to that of PETA and SAFE. The result of the constant negative media can seriously damage the reputation, image and popularity of an organisation<ref name= "Bruce"/>. This then has an effect on areas that involve attendance, merchandising, sponsorship and endorsement deals. When expressing their opinions on animal cruelty organisations are discouraging people from attending events which could effect the ability for sporting businesses to make a profit<ref name= "Bruce"/>. An example of when sponsorship was removed is when Telstra removed sponsorship from a local rodeo in Corryong due to allegations of animal cruelty<ref>ABC News, ‘Telstra bucks sponsorship over rodeo cruelty fears’, in ABC News, September 2007, viewed on 4 September 2011, < http://www.abc.net.au/news/2007-09-19/telstra-bucks-sponsorship-over-rodeo-cruelty-fears/674208></ref>. . Another instance when sponsorship was removes was when Choice Hotels dropped their sponsorship of the National High School Rodeo Association (NHRSA) because an animal rights group alleging that by sponsoring the event they were supporting animal cruelty <ref name = "SHARK"/>.The Loss of sponsorship has an affect on sporting businesses as they are losing funds that are essential to hold and run events. Therefore the way that the organisation is portrayed in the media can have major impacts on sporting businesses. Sponsorship of an event is supposed to have a positive effect on the corporate image of an organisation <ref name= "Wilson">Wilson, Bradley, Stavros, Constantino & Westburg, Kate ‘A sport crisis typology: establishing a pathway for future research’, International Journal of Sport Management & marketing, vol. 7, no. 1, 2010, p 21</ref>. The association of a brand with sport can have negative effect if the sporting organisation generates a lot of negative media attention. This has the potential cause trouble for the sports that are involved in competitions which involve animal cruelty claims. Sponsorship of these events could be viewed as risk taking behaviour <ref name="Wilson"/>. Sponsorship is a fundamental ingredient for a successful event. It is a major stream of revenue for the organisers of an event and the withdrawal of sponsorship can have major impacts on an event and cause financial difficulties for organisations<ref>Westburg, Kate, Stavros, Kate & Wilson, Bradley ‘The impact of degenerative episodes on the sponsorship B2B relationship: Implications for brand management’ Journal of Industrial Marketing Management, vol 40, 2011, pp 603-611</ref>. There was controversy amongst the Greyhound racing industry when apparently not one representative from the McGrath Foundation turned up to receive a check from the greyhound racing industry. This was due to the public perceptions that by being associated with this organisation they support animal cruelty. However, they still wanted to receive the money, they just did not want to do it in the public eye <ref>Animals Australia, ‘The Australian Greyhound Racing Industry: The McGrath Foundation’, in Unleashed. March 2011, viewed 4 September 2011, <http://www.unleashed.org.au/community/forum/topic.php?t=4724></ref>. This affects the business of greyhound racing as people do not want to be involved or associated with sports that attract negative publicity as they do not want to be seen as supporting animal cruelty. Many other concerns arise as animal rights activists believe that hitting the horse with a whip is considered to be animal cruelty<ref name="ARB"/>. These allegations could possibly have an effect on the commodification of this sport as the activists discourage people from attending these events or placing bets <ref>Animals Australia, ‘the glitz the glam our... the grim reality’ in Horse Racing. August 2010, viewed 5 September 2011, < http://www.animalsaustralia.org/issues/horse_racing.php#toc4></ref>. They convince people that if they attend an event they are supporting animal cruelty, this is the same as in Rodeo. The affect of the constant scrutiny resulted in the ARB changing the rules of racing as these issues were raised at all levels of competition. Jockeys are no longer aloud to raise whips above their shoulder, also whips must be padded <ref name= "ARB"> Australian Racing Board, ‘Whip Reform Stays’, in Australian Racing Board. September 2009, viewed on 1 September 2011, <http://www.australianracingboard.com.au/Media.whip-reform-stays></ref>. Within 5 weeks of changing the rules the ARB believed that there had been a substantial improvement in the attitudes and practices of this sport <ref name="ARB"/>. This has had a positive impact on the horse racing industry. However, the opinions of the media still effect areas such as sponsorship for sporting organisations. {{center top}} {| border=1 cellspacing=5 cellpadding=5 |- | [[File:Horse-racing-1.jpg|thumb|300px|center|Horse racing at Galopp Riem 05/06/2005, Munich, Germany.Photo by Softeis.]] | [[File:Greyhound Racing 2 amk.jpg|thumb|350px|center|English: Greyhound racing 18/05/2008.Phot by AngMokio]] |} {{center bottom}} ==How Organisations Respond to Such Allegations == In defence to the animal cruelty claims the APRA argue that there is no information to support the claims that animals are treated in an inhumane way. They also believe that the statements that are made by groups such as PETA and SAFE are fabricated to support their personal emotions regarding the sport<ref name="APRA"/>. In repsonsding to animal cruelty claims in rodeo the APRA said that "The sport has had to become more transparent and proactive if it was to survive under today’s expectations"<ref name= "Stephens">Stephens, Rhiannon, 'Email interview transcript with Australian Professional Rodeo Association and Greyhound Racing Australia' in User_talk:Rhiannon Stephens. 2011, <http://en.wikiversity.org/wiki/User_talk:Rhiannon_Stephens/The_Effect_of_Animal_Cruelty_Allegations_in_Sport></ref> Wikiversity user:Rhiannon Stephens talk page</ref>. Statistics of injured livestock are taken at events and forwarded on to animal welfare regulatory bodies so that they can compare them with their own records. The APRA have also stated on their website that they value the animals that they use and follow strict animal welfare policies <ref name="APRA"/>. This website also explains for people that do not understand the sport of Rodeo that the use of these animals in such a way does not involve animal cruelty<ref>Australian Professional Rodeo Association, ‘The Facts Concerning the Care and Treatment of Professional Rodeo Livestock’, in Animal Welfare .n.d, viewed on 5 September 2011, <http://prorodeo.asn.au/forms/AnimalWelfare.pdf></ref>. As seen in the horse racing example the ARB responded to animal cruelty allegations by changing the rules and regulations regarding whips. They believe that by having done this they have seen an improvement in the attitudes of people around the sport, therefore this has had a positive impact. In order to prevent allegations GRSA have partnered with Technical and Further Education (TAFE) and the University of South Australia to ensure that their animals receive the best possible care(Email). Also the dogs are placed in foster homes so that they learn how to become pets and once they are finished racing are placed into permanent homes<ref name= "Stephens"/>. They also ensure people that they put animal welfare at the heart of what they do<ref name ="Stephens"/>. All organisations that are involved in these sports have made sure that policies in place to protect the animals that are involved. ==Conclusion== Animals have been used in sports since the ancient Greek and Roman chariot races, and continue to be used in professional sports today<ref name= "American"/>. The emergence of animals in sport brought about animal welfare legislation in the 1870’s, which provides the laws for which animals are to be treated<ref name= "Graves"/>. These sports are a major source of entertainment and provide huge media and sponsorship deals. The organisations are not able to control the opinions that are expressed in the media <ref name= "Bruce"/>. The negative publicity that is generated has huge impacts on sporting organisations<ref name= "Bruce"/>. In the media, animal rights organisations claim that the use of animals in sport is what constitutes animal cruelty and we should have total animal liberation. In stating their opinions the animal rights organisations discourage people from attending events and betting. These articles are negative and people are more likely to remember the facts from these than positive articles<ref name="Funk"/>. The negative media coverage damages the reputation, image and popularity of an organisation which in turn affects attendance, merchandising, sponsorship and endorsement deals as other people and organisations do not want to be seen as supporting animal cruelty<ref name= "Bruce"/>. This is a major concern where sponsorship is concerned as association with an organisation can have a negative effect if it attracts substantial amounts of negative publicity, in which these sports do<ref name="Bruce"/>. The statements that are made in the media to the public have a huge impact on the businesses that are involved and this then affects all aspects of the sport<ref name= "Bruce"/>. The organisations involved claim that there is no information to support the claims made by animal rights organisations that their opinions are fabricated to represent their personal emotions regarding the sport. ==References== {{reflist}} {{CourseCat}} h14yvfqsndibgcps73clc2ceiaqr2p2 Portal:Filipino 102 137916 2803495 1667183 2026-04-08T07:00:14Z CarlessParking 3064444 2803495 wikitext text/x-wiki Welcome to the Filipino Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. == Introduction == Filipino is a prestige register of the Tagalog language, and the name under which Tagalog is designated the national language of the Philippines, as well as an official language alongside English. Tagalog is a first language of about one-third of the Philippine population; it is centered around Manila but is spoken to varying degrees nationwide. Filipino is constitutionally designated as the national language of the Philippines and, along with English, one of two official languages. == Courses/Projects == == Resources == == Vocabulary == == Grammar == == Division News == * '''02 November 2012 ''' - Department founded! == Things You Can Do == == Wikipedia == 1. [[Wikipedia:Filipino language|Filipino Language]] == External Resources == [[Category:Southeast Asian languages|Southeast Asian Language]][[Category:Filipino]] [[Category:Language learning portals]] hfdh8b73bpneo04vpbuyowy886cu2m9 Understanding Arithmetic Circuits 0 139384 2803327 2803237 2026-04-07T13:37:23Z Young1lim 21186 /* Adder */ 2803327 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260407.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] 7ksulp8kufj1rz43fdifva1bi4uzjh9 Lua/Tables 0 153528 2803384 2803268 2026-04-07T19:00:21Z Nimmzo 801528 /* Create a Lua Script with Tables */ -ending comma +syntaxhighlight: +inline. /* Understand Your Lua Script */ + /* tableToString */ parameter t → tab + /* p.sequence */ + /* p.dictionary */ 2803384 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a [[mw:Extension:Scribunto/Lua_reference_manual#table|table]] to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-06}}</ref> Tables are [[w:en:associative array|associative arrays]] or collections of data with key / value pairs that may be used to hold, organize and access data. This lesson will show you how to use tables in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Functions | Functions]] lesson. == Console to learn Lua == # In [[Module:Sandbox]], click <code>Edit source</code> without modifying the existing code. # Scroll down until <code>Debug console</code> at the end of the page. # Paste the following entire function and its call together then validate once by Enter: ↲ <blockquote><syntaxhighlight lang="lua" line highlight=1,14 copy> local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot 𝛼 end end return table.concat(res) -- concatenation more efficient than .. operator end local lstNum = {10, 20, 30} print(tableToString(lstNum)) </syntaxhighlight></blockquote> The result should be: <blockquote><syntaxhighlight lang="lua" line start=15> :table[1] is 10. table[2] is 20. table[3] is 30. </syntaxhighlight></blockquote> Let's define a dictionary of Asian languages: <blockquote><syntaxhighlight lang="lua" line highlight=1,3,4 start=16 copy> local lstAsian = {ja = "日本語", ko = "한국어", zh = "中文"} print(lstNum[2]) print(lstAsian.ja) print(tableToString(lstAsian)) </syntaxhighlight></blockquote> The Debug console supports Unicode: <blockquote><syntaxhighlight lang="lua" line start=20> 20 日本語 :table.ja is 日本語. table.zh is 中文. table.ko is 한국어. </syntaxhighlight></blockquote> CAUTION: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order. Note: press upwards arrow ↑ for the historic in the Debug console. == Create a Lua Script with Tables == To create a Lua script with tables: # Go to the top of [[Module:Sandbox]] in the editor. # Replace the existing code pasting the following Lua code without clicking on <code>Publish changes</code>. <blockquote><syntaxhighlight lang="lua" line highlight=2,14,15,21 copy> local p = {} local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot alpha end end return table.concat(res) -- concatenation more efficient than string `..` operator end function p.sequence() return ';sequence\n' .. tableToString({10, 20, 30}) end -- numerical sequence function p.dictionary() -- returns a dictionary of language codes -> English names local lstLang = {de = "German", en = "English", es = "Spanish", fr = "French", it = "Italian", -- Europe ja = "Japanese", ko = "Korean", ru = "Russian", zh = "Chinese" -- Asia + Russia } return ';dictionary\n' .. tableToString(lstLang) end function p.translate() -- provides language codes -> localized native names local lstLang = {de = "Deutsch", en = "English", es = "Español", fr = "Français", it = "Italiano", -- Latin ja = "日本語", ko = "한국어", ru = "Русский", zh = "中文" -- Unicode } return ';translate\n' .. tableToString(lstLang) end return p </syntaxhighlight></blockquote> In <code>Debug console</code>, validate: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> print(p.sequence()) print(p.dictionary()) print(p.translate()) </syntaxhighlight></blockquote> WARNING: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order: <blockquote><syntaxhighlight lang="lua" line start=4> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Save the Module:Sandbox to call it from another Wiki page. # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> {{#invoke:Sandbox|sequence}} {{#invoke:Sandbox|dictionary}} {{#invoke:Sandbox|translate}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </blockquote> == Understand Your Lua Script == === tableToString === To understand your Lua script <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function tableToString(tab)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>, which is the table to be converted to a string. # <syntaxhighlight lang="lua" inline>local</syntaxhighlight> and the following code defines the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight>, <syntaxhighlight lang="lua" inline>val</syntaxhighlight>, and <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. All are <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # The previous lesson is based on string <syntaxhighlight lang="lua" inline>result = ''</syntaxhighlight>.<br>Let's introduce the table <syntaxhighlight lang="lua" inline>local res = {':'} -- left margin</syntaxhighlight> with its counter <syntaxhighlight lang="lua" inline>#res -- length of the table</syntaxhighlight> # <syntaxhighlight lang="lua" inline>for key, value in pairs(t) do</syntaxhighlight> creates a loop code block that will vary the value of the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight> and <syntaxhighlight lang="lua" inline>value</syntaxhighlight> for each key/value data pair in the table <syntaxhighlight lang="lua" inline>t</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>if (tonumber(key) ~= nil) then</syntaxhighlight> tests to see if the key can be converted to a number. If it can, the key is displayed as a number (without quotes). If it can't be converted, the key is displayed as a literal string (with quotes). # Compare the string concatenation: <syntaxhighlight lang="lua" inline>result = result .. ':table[' .. key .. '] is ' .. value .. '\n'</syntaxhighlight> or <syntaxhighlight lang="lua" inline>result = result .. ':table[\'' .. key .. '\'] is ' .. value .. '\n'</syntaxhighlight> adds the current key and value to the result.<br><syntaxhighlight lang="lua" inline>res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index]</syntaxhighlight> will be faster thanks to <syntaxhighlight lang="lua" inline>return table.concat(res) -- concatenation more efficient than string `..` operator</syntaxhighlight>. #* The <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation is a common way to reference individual table keys to access their associated values. #* To display quotes inside a string, you must either switch between quote identifiers ("/'), or use the \ character to 'escape' the quote, which forces it to be part of the string rather than terminate the string. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the loop. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.sequence === To understand your Lua script <syntaxhighlight lang="lua" inline>sequence</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.sequence()</syntaxhighlight> declares a function named sequence. # <syntaxhighlight lang="lua" inline>local numbers = {10, 20, 30}</syntaxhighlight> defines a local table variable named <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight>, which is initialized with three values. #: When table values are specified without matching keys, Lua automatically adds numeric keys for the values as a sequence from 1 to N, the number of values added. This allows Lua tables to be used similar to how arrays are used in other programming languages. For example, the second value in the table could be referenced as <syntaxhighlight lang="lua" inline>numbers[2]</syntaxhighlight>, and the table values could be processed with a for loop. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';sequence\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. The semi colon is an operator that separates each new statement and the /n means new line. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(numbers)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.dictionary === To understand your Lua script <syntaxhighlight lang="lua" inline>dictionary</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.dictionary()</syntaxhighlight> declares a function named dictionary. # <syntaxhighlight lang="lua" inline>local languages = {</syntaxhighlight> and the following code defines a local table variable named <syntaxhighlight lang="lua" inline>languages</syntaxhighlight>, which is initialized with 9 key/value pairs. #* When table values are specified with matching keys, Lua uses the specified keys instead of numeric key values. #* Lua tables with specified key values cannot be used as arrays using <syntaxhighlight lang="lua" inline>table[number]</syntaxhighlight> notation. #* Lua tables with specified key values can be used as associative arrays using <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>languages['en']</syntaxhighlight>. Quotes around string literal keys are required with this notation format. #* Lua tables with specified key values can also be used as associative arrays using <syntaxhighlight lang="lua" inline>table.key</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>languages.en</syntaxhighlight>. Quotes around string literal keys are not required with this notation format. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';dictionary\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(languages)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>languages</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. It is important to note that the for loop to process table pairs will process all values in the table, but tables with specified key values may not be processed in the order they were created. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with tables. Continue on to the [[../Errors/]] lesson. == See Also == * [[Lua/Table Library | Lua Table Library]] * [[Wikipedia: Associative array]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] 6gr0h7kmvg9wfltt8kmihkjrc5x3243 2803388 2803384 2026-04-07T19:20:51Z Nimmzo 801528 /* p.dictionary or p.translate */ languages → lstLang shorter to make room for the ending -- comment 2803388 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a [[mw:Extension:Scribunto/Lua_reference_manual#table|table]] to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-06}}</ref> Tables are [[w:en:associative array|associative arrays]] or collections of data with key / value pairs that may be used to hold, organize and access data. This lesson will show you how to use tables in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Functions | Functions]] lesson. == Console to learn Lua == # In [[Module:Sandbox]], click <code>Edit source</code> without modifying the existing code. # Scroll down until <code>Debug console</code> at the end of the page. # Paste the following entire function and its call together then validate once by Enter: ↲ <blockquote><syntaxhighlight lang="lua" line highlight=1,14 copy> local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot 𝛼 end end return table.concat(res) -- concatenation more efficient than .. operator end local lstNum = {10, 20, 30} print(tableToString(lstNum)) </syntaxhighlight></blockquote> The result should be: <blockquote><syntaxhighlight lang="lua" line start=15> :table[1] is 10. table[2] is 20. table[3] is 30. </syntaxhighlight></blockquote> Let's define a dictionary of Asian languages: <blockquote><syntaxhighlight lang="lua" line highlight=1,3,4 start=16 copy> local lstAsian = {ja = "日本語", ko = "한국어", zh = "中文"} print(lstNum[2]) print(lstAsian.ja) print(tableToString(lstAsian)) </syntaxhighlight></blockquote> The Debug console supports Unicode: <blockquote><syntaxhighlight lang="lua" line start=20> 20 日本語 :table.ja is 日本語. table.zh is 中文. table.ko is 한국어. </syntaxhighlight></blockquote> CAUTION: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order. Note: press upwards arrow ↑ for the historic in the Debug console. == Create a Lua Script with Tables == To create a Lua script with tables: # Go to the top of [[Module:Sandbox]] in the editor. # Replace the existing code pasting the following Lua code without clicking on <code>Publish changes</code>. <blockquote><syntaxhighlight lang="lua" line highlight=2,14,15,21 copy> local p = {} local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot alpha end end return table.concat(res) -- concatenation more efficient than string `..` operator end function p.sequence() return ';sequence\n' .. tableToString({10, 20, 30}) end -- numerical sequence function p.dictionary() -- returns a dictionary of language codes -> English names local lstLang = {de = "German", en = "English", es = "Spanish", fr = "French", it = "Italian", -- Europe ja = "Japanese", ko = "Korean", ru = "Russian", zh = "Chinese" -- Asia + Russia } return ';dictionary\n' .. tableToString(lstLang) end function p.translate() -- provides language codes -> localized native names local lstLang = {de = "Deutsch", en = "English", es = "Español", fr = "Français", it = "Italiano", -- Latin ja = "日本語", ko = "한국어", ru = "Русский", zh = "中文" -- Unicode } return ';translate\n' .. tableToString(lstLang) end return p </syntaxhighlight></blockquote> In <code>Debug console</code>, validate: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> print(p.sequence()) print(p.dictionary()) print(p.translate()) </syntaxhighlight></blockquote> WARNING: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order: <blockquote><syntaxhighlight lang="lua" line start=4> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Save the Module:Sandbox to call it from another Wiki page. # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> {{#invoke:Sandbox|sequence}} {{#invoke:Sandbox|dictionary}} {{#invoke:Sandbox|translate}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </blockquote> == Understand Your Lua Script == === tableToString === To understand your Lua script <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function tableToString(tab)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>, which is the table to be converted to a string. # <syntaxhighlight lang="lua" inline>local</syntaxhighlight> and the following code defines the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight>, <syntaxhighlight lang="lua" inline>val</syntaxhighlight>, and <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. All are <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # The previous lesson is based on string <syntaxhighlight lang="lua" inline>result = ''</syntaxhighlight>.<br>Let's introduce the table <syntaxhighlight lang="lua" inline>local res = {':'} -- left margin</syntaxhighlight> with its counter <syntaxhighlight lang="lua" inline>#res -- length of the table</syntaxhighlight> # <syntaxhighlight lang="lua" inline>for key, value in pairs(t) do</syntaxhighlight> creates a loop code block that will vary the value of the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight> and <syntaxhighlight lang="lua" inline>value</syntaxhighlight> for each key/value data pair in the table <syntaxhighlight lang="lua" inline>t</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>if (tonumber(key) ~= nil) then</syntaxhighlight> tests to see if the key can be converted to a number. If it can, the key is displayed as a number (without quotes). If it can't be converted, the key is displayed as a literal string (with quotes). # Compare the string concatenation: <syntaxhighlight lang="lua" inline>result = result .. ':table[' .. key .. '] is ' .. value .. '\n'</syntaxhighlight> or <syntaxhighlight lang="lua" inline>result = result .. ':table[\'' .. key .. '\'] is ' .. value .. '\n'</syntaxhighlight> adds the current key and value to the result.<br><syntaxhighlight lang="lua" inline>res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index]</syntaxhighlight> will be faster thanks to <syntaxhighlight lang="lua" inline>return table.concat(res) -- concatenation more efficient than string `..` operator</syntaxhighlight>. #* The <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation is a common way to reference individual table keys to access their associated values. #* To display quotes inside a string, you must either switch between quote identifiers ("/'), or use the \ character to 'escape' the quote, which forces it to be part of the string rather than terminate the string. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the loop. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.sequence === To understand your Lua script <syntaxhighlight lang="lua" inline>sequence</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.sequence()</syntaxhighlight> declares a function named sequence. # <syntaxhighlight lang="lua" inline>local numbers = {10, 20, 30}</syntaxhighlight> defines a local table variable named <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight>, which is initialized with three values. #: When table values are specified without matching keys, Lua automatically adds numeric keys for the values as a sequence from 1 to N, the number of values added. This allows Lua tables to be used similar to how arrays are used in other programming languages. For example, the second value in the table could be referenced as <syntaxhighlight lang="lua" inline>numbers[2]</syntaxhighlight>, and the table values could be processed with a for loop. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';sequence\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. The semi colon is an operator that separates each new statement and the /n means new line. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(numbers)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.dictionary or p.translate === To understand your Lua script <syntaxhighlight lang="lua" inline>dictionary</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.dictionary()</syntaxhighlight> declares a function named dictionary. # <syntaxhighlight lang="lua" inline>local lstLang = {de = "German", en = "English",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>ja = "日本語", ko = "한국어",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>, zh = "中文"} -- Unicode</syntaxhighlight> and the following code defines a local table variable named <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight>, which is initialized with 9 key/value pairs. #* When table values are specified with matching keys, Lua uses the specified keys instead of numeric key values. #* Lua tables with specified key values cannot be used as arrays using <syntaxhighlight lang="lua" inline>table[number]</syntaxhighlight> notation. #* Lua tables with specified key values can be used as associative arrays using <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang['ja']</syntaxhighlight>. Quotes around string literal keys are required with this notation format. #* Lua tables with specified key values can also be used as associative arrays using <syntaxhighlight lang="lua" inline>table.key</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang.ja</syntaxhighlight>. Quotes around string literal keys are not required with this notation format. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';dictionary\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(languages)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>languages</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. It is important to note that the for loop to process table pairs will process all values in the table, but tables with specified key values may not be processed in the order they were created. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with tables. Continue on to the [[../Errors/]] lesson. == See Also == * [[Lua/Table Library | Lua Table Library]] * [[Wikipedia: Associative array]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] 2mwg9lb2rf8f2pkfmfqlibmlffcfe5t 2803390 2803388 2026-04-07T19:40:18Z Nimmzo 801528 /* p.dictionary or p.translate */ simplify return -result 2803390 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a [[mw:Extension:Scribunto/Lua_reference_manual#table|table]] to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-06}}</ref> Tables are [[w:en:associative array|associative arrays]] or collections of data with key / value pairs that may be used to hold, organize and access data. This lesson will show you how to use tables in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Functions | Functions]] lesson. == Console to learn Lua == # In [[Module:Sandbox]], click <code>Edit source</code> without modifying the existing code. # Scroll down until <code>Debug console</code> at the end of the page. # Paste the following entire function and its call together then validate once by Enter: ↲ <blockquote><syntaxhighlight lang="lua" line highlight=1,14 copy> local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot 𝛼 end end return table.concat(res) -- concatenation more efficient than .. operator end local lstNum = {10, 20, 30} print(tableToString(lstNum)) </syntaxhighlight></blockquote> The result should be: <blockquote><syntaxhighlight lang="lua" line start=15> :table[1] is 10. table[2] is 20. table[3] is 30. </syntaxhighlight></blockquote> Let's define a dictionary of Asian languages: <blockquote><syntaxhighlight lang="lua" line highlight=1,3,4 start=16 copy> local lstAsian = {ja = "日本語", ko = "한국어", zh = "中文"} print(lstNum[2]) print(lstAsian.ja) print(tableToString(lstAsian)) </syntaxhighlight></blockquote> The Debug console supports Unicode: <blockquote><syntaxhighlight lang="lua" line start=20> 20 日本語 :table.ja is 日本語. table.zh is 中文. table.ko is 한국어. </syntaxhighlight></blockquote> CAUTION: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order. Note: press upwards arrow ↑ for the historic in the Debug console. == Create a Lua Script with Tables == To create a Lua script with tables: # Go to the top of [[Module:Sandbox]] in the editor. # Replace the existing code pasting the following Lua code without clicking on <code>Publish changes</code>. <blockquote><syntaxhighlight lang="lua" line highlight=2,14,15,21 copy> local p = {} local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot alpha end end return table.concat(res) -- concatenation more efficient than string `..` operator end function p.sequence() return ';sequence\n' .. tableToString({10, 20, 30}) end -- numerical sequence function p.dictionary() -- returns a dictionary of language codes -> English names local lstLang = {de = "German", en = "English", es = "Spanish", fr = "French", it = "Italian", -- Europe ja = "Japanese", ko = "Korean", ru = "Russian", zh = "Chinese" -- Asia + Russia } return ';dictionary\n' .. tableToString(lstLang) end function p.translate() -- provides language codes -> localized native names local lstLang = {de = "Deutsch", en = "English", es = "Español", fr = "Français", it = "Italiano", -- Latin ja = "日本語", ko = "한국어", ru = "Русский", zh = "中文" -- Unicode } return ';translate\n' .. tableToString(lstLang) end return p </syntaxhighlight></blockquote> In <code>Debug console</code>, validate: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> print(p.sequence()) print(p.dictionary()) print(p.translate()) </syntaxhighlight></blockquote> WARNING: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order: <blockquote><syntaxhighlight lang="lua" line start=4> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Save the Module:Sandbox to call it from another Wiki page. # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> {{#invoke:Sandbox|sequence}} {{#invoke:Sandbox|dictionary}} {{#invoke:Sandbox|translate}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </blockquote> == Understand Your Lua Script == === tableToString === To understand your Lua script <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function tableToString(tab)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>, which is the table to be converted to a string. # <syntaxhighlight lang="lua" inline>local</syntaxhighlight> and the following code defines the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight>, <syntaxhighlight lang="lua" inline>val</syntaxhighlight>, and <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. All are <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # The previous lesson is based on string <syntaxhighlight lang="lua" inline>result = ''</syntaxhighlight>.<br>Let's introduce the table <syntaxhighlight lang="lua" inline>local res = {':'} -- left margin</syntaxhighlight> with its counter <syntaxhighlight lang="lua" inline>#res -- length of the table</syntaxhighlight> # <syntaxhighlight lang="lua" inline>for key, value in pairs(t) do</syntaxhighlight> creates a loop code block that will vary the value of the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight> and <syntaxhighlight lang="lua" inline>value</syntaxhighlight> for each key/value data pair in the table <syntaxhighlight lang="lua" inline>t</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>if (tonumber(key) ~= nil) then</syntaxhighlight> tests to see if the key can be converted to a number. If it can, the key is displayed as a number (without quotes). If it can't be converted, the key is displayed as a literal string (with quotes). # Compare the string concatenation: <syntaxhighlight lang="lua" inline>result = result .. ':table[' .. key .. '] is ' .. value .. '\n'</syntaxhighlight> or <syntaxhighlight lang="lua" inline>result = result .. ':table[\'' .. key .. '\'] is ' .. value .. '\n'</syntaxhighlight> adds the current key and value to the result.<br><syntaxhighlight lang="lua" inline>res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index]</syntaxhighlight> will be faster thanks to <syntaxhighlight lang="lua" inline>return table.concat(res) -- concatenation more efficient than string `..` operator</syntaxhighlight>. #* The <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation is a common way to reference individual table keys to access their associated values. #* To display quotes inside a string, you must either switch between quote identifiers ("/'), or use the \ character to 'escape' the quote, which forces it to be part of the string rather than terminate the string. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the loop. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.sequence === To understand your Lua script <syntaxhighlight lang="lua" inline>sequence</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.sequence()</syntaxhighlight> declares a function named sequence. # <syntaxhighlight lang="lua" inline>local numbers = {10, 20, 30}</syntaxhighlight> defines a local table variable named <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight>, which is initialized with three values. #: When table values are specified without matching keys, Lua automatically adds numeric keys for the values as a sequence from 1 to N, the number of values added. This allows Lua tables to be used similar to how arrays are used in other programming languages. For example, the second value in the table could be referenced as <syntaxhighlight lang="lua" inline>numbers[2]</syntaxhighlight>, and the table values could be processed with a for loop. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';sequence\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. The semi colon is an operator that separates each new statement and the /n means new line. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(numbers)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.dictionary or p.translate === To understand your Lua script <syntaxhighlight lang="lua" inline>dictionary</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.dictionary()</syntaxhighlight> declares a function named dictionary. # <syntaxhighlight lang="lua" inline>local lstLang = {de = "German", en = "English",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>ja = "日本語", ko = "한국어",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>, zh = "中文"} -- Unicode</syntaxhighlight> and the following code defines a local table variable named <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight>, which is initialized with 9 key/value pairs. #* When table values are specified with matching keys, Lua uses the specified keys instead of numeric key values. #* Lua tables with specified key values cannot be used as arrays using <syntaxhighlight lang="lua" inline>table[number]</syntaxhighlight> notation. #* Lua tables with specified key values can be used as associative arrays using <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang['ja']</syntaxhighlight>. Quotes around string literal keys are required with this notation format. #* Lua tables with specified key values can also be used as associative arrays using <syntaxhighlight lang="lua" inline>table.key</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang.ja</syntaxhighlight>. Quotes around string literal keys are not required with this notation format. # <syntaxhighlight lang="lua" inline>return ';dictionary\n' .. tableToString(lstLang)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight> as the parameter and concatenates the returned value as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. It is important to note that the for loop to process table pairs will process all values in the table, but tables with specified key values may not be processed in the order they were created. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with tables. Continue on to the [[../Errors/]] lesson. == See Also == * [[Lua/Table Library | Lua Table Library]] * [[Wikipedia: Associative array]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] 7me051f7qtrfwhsdtqc3ll24hhga3t8 2803393 2803390 2026-04-07T19:50:23Z Nimmzo 801528 /* tableToString */ update return 2803393 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a [[mw:Extension:Scribunto/Lua_reference_manual#table|table]] to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-06}}</ref> Tables are [[w:en:associative array|associative arrays]] or collections of data with key / value pairs that may be used to hold, organize and access data. This lesson will show you how to use tables in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Functions | Functions]] lesson. == Console to learn Lua == # In [[Module:Sandbox]], click <code>Edit source</code> without modifying the existing code. # Scroll down until <code>Debug console</code> at the end of the page. # Paste the following entire function and its call together then validate once by Enter: ↲ <blockquote><syntaxhighlight lang="lua" line highlight=1,14 copy> local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot 𝛼 end end return table.concat(res) -- concatenation more efficient than .. operator end local lstNum = {10, 20, 30} print(tableToString(lstNum)) </syntaxhighlight></blockquote> The result should be: <blockquote><syntaxhighlight lang="lua" line start=15> :table[1] is 10. table[2] is 20. table[3] is 30. </syntaxhighlight></blockquote> Let's define a dictionary of Asian languages: <blockquote><syntaxhighlight lang="lua" line highlight=1,3,4 start=16 copy> local lstAsian = {ja = "日本語", ko = "한국어", zh = "中文"} print(lstNum[2]) print(lstAsian.ja) print(tableToString(lstAsian)) </syntaxhighlight></blockquote> The Debug console supports Unicode: <blockquote><syntaxhighlight lang="lua" line start=20> 20 日本語 :table.ja is 日本語. table.zh is 中文. table.ko is 한국어. </syntaxhighlight></blockquote> CAUTION: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order. Note: press upwards arrow ↑ for the historic in the Debug console. == Create a Lua Script with Tables == To create a Lua script with tables: # Go to the top of [[Module:Sandbox]] in the editor. # Replace the existing code pasting the following Lua code without clicking on <code>Publish changes</code>. <blockquote><syntaxhighlight lang="lua" line highlight=2,14,15,21 copy> local p = {} local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot alpha end end return table.concat(res) -- concatenation more efficient than string `..` operator end function p.sequence() return ';sequence\n' .. tableToString({10, 20, 30}) end -- numerical sequence function p.dictionary() -- returns a dictionary of language codes -> English names local lstLang = {de = "German", en = "English", es = "Spanish", fr = "French", it = "Italian", -- Europe ja = "Japanese", ko = "Korean", ru = "Russian", zh = "Chinese" -- Asia + Russia } return ';dictionary\n' .. tableToString(lstLang) end function p.translate() -- provides language codes -> localized native names local lstLang = {de = "Deutsch", en = "English", es = "Español", fr = "Français", it = "Italiano", -- Latin ja = "日本語", ko = "한국어", ru = "Русский", zh = "中文" -- Unicode } return ';translate\n' .. tableToString(lstLang) end return p </syntaxhighlight></blockquote> In <code>Debug console</code>, validate: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> print(p.sequence()) print(p.dictionary()) print(p.translate()) </syntaxhighlight></blockquote> WARNING: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order: <blockquote><syntaxhighlight lang="lua" line start=4> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Save the Module:Sandbox to call it from another Wiki page. # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> {{#invoke:Sandbox|sequence}} {{#invoke:Sandbox|dictionary}} {{#invoke:Sandbox|translate}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </blockquote> == Understand Your Lua Script == === tableToString === To understand your Lua script <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function tableToString(tab)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>, which is the table to be converted to a string. # <syntaxhighlight lang="lua" inline>local</syntaxhighlight> and the following code defines the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight>, <syntaxhighlight lang="lua" inline>val</syntaxhighlight>, and <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. All are <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # The previous lesson is based on string <syntaxhighlight lang="lua" inline>result = ''</syntaxhighlight>.<br>Let's introduce the table <syntaxhighlight lang="lua" inline>local res = {':'} -- left margin</syntaxhighlight> with its counter <syntaxhighlight lang="lua" inline>#res -- length of the table</syntaxhighlight> # <syntaxhighlight lang="lua" inline>for key, value in pairs(tab) do</syntaxhighlight> creates a loop code block that will vary the value of the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight> and <syntaxhighlight lang="lua" inline>value</syntaxhighlight> for each key/value data pair in the table <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>if tonumber(key) then</syntaxhighlight> tests to see if the key can be converted to a number. If it can, the key is displayed as a number (without quotes). If it can't be converted, the key is displayed as a literal string (with quotes). # Compare the string concatenation: <syntaxhighlight lang="lua" inline>result = result .. ':table[' .. key .. '] is ' .. value .. '\n'</syntaxhighlight> or<br><syntaxhighlight lang="lua" inline>result = result .. ':table[\'' .. key .. '\'] is ' .. value .. '\n'</syntaxhighlight> adds the current key and value to the result.<br><syntaxhighlight lang="lua" inline>res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index]</syntaxhighlight> will be faster thanks to <syntaxhighlight lang="lua" inline>return table.concat(res)</syntaxhighlight>. #* The <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation is a common way to reference individual table keys to access their associated values. #* To display quotes inside a string, you must either switch between quote identifiers ("/'), or use the \ character to 'escape' the quote, which forces it to be part of the string rather than terminate the string. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the loop. # <syntaxhighlight lang="lua" inline>return table.concat(res) -- concatenation more efficient than string `..` operator</syntaxhighlight> returns the concatenation of each item in the table as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.sequence === To understand your Lua script <syntaxhighlight lang="lua" inline>sequence</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.sequence()</syntaxhighlight> declares a function named sequence. # <syntaxhighlight lang="lua" inline>local numbers = {10, 20, 30}</syntaxhighlight> defines a local table variable named <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight>, which is initialized with three values. #: When table values are specified without matching keys, Lua automatically adds numeric keys for the values as a sequence from 1 to N, the number of values added. This allows Lua tables to be used similar to how arrays are used in other programming languages. For example, the second value in the table could be referenced as <syntaxhighlight lang="lua" inline>numbers[2]</syntaxhighlight>, and the table values could be processed with a for loop. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';sequence\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. The semi colon is an operator that separates each new statement and the /n means new line. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(numbers)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.dictionary or p.translate === To understand your Lua script <syntaxhighlight lang="lua" inline>dictionary</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.dictionary()</syntaxhighlight> declares a function named dictionary. # <syntaxhighlight lang="lua" inline>local lstLang = {de = "German", en = "English",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>ja = "日本語", ko = "한국어",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>, zh = "中文"} -- Unicode</syntaxhighlight> and the following code defines a local table variable named <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight>, which is initialized with 9 key/value pairs. #* When table values are specified with matching keys, Lua uses the specified keys instead of numeric key values. #* Lua tables with specified key values cannot be used as arrays using <syntaxhighlight lang="lua" inline>table[number]</syntaxhighlight> notation. #* Lua tables with specified key values can be used as associative arrays using <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang['ja']</syntaxhighlight>. Quotes around string literal keys are required with this notation format. #* Lua tables with specified key values can also be used as associative arrays using <syntaxhighlight lang="lua" inline>table.key</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang.ja</syntaxhighlight>. Quotes around string literal keys are not required with this notation format. # <syntaxhighlight lang="lua" inline>return ';dictionary\n' .. tableToString(lstLang)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight> as the parameter and concatenates the returned value as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. It is important to note that the for loop to process table pairs will process all values in the table, but tables with specified key values may not be processed in the order they were created. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with tables. Continue on to the [[../Errors/]] lesson. == See Also == * [[Lua/Table Library | Lua Table Library]] * [[Wikipedia: Associative array]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] ppy63855umzx7u9100yt9pyt187cegx 2803398 2803393 2026-04-07T20:10:11Z Nimmzo 801528 /* p.sequence */ compare the short version with the detailed lines 2803398 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a [[mw:Extension:Scribunto/Lua_reference_manual#table|table]] to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-06}}</ref> Tables are [[w:en:associative array|associative arrays]] or collections of data with key / value pairs that may be used to hold, organize and access data. This lesson will show you how to use tables in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Functions | Functions]] lesson. == Console to learn Lua == # In [[Module:Sandbox]], click <code>Edit source</code> without modifying the existing code. # Scroll down until <code>Debug console</code> at the end of the page. # Paste the following entire function and its call together then validate once by Enter: ↲ <blockquote><syntaxhighlight lang="lua" line highlight=1,14 copy> local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot 𝛼 end end return table.concat(res) -- concatenation more efficient than .. operator end local lstNum = {10, 20, 30} print(tableToString(lstNum)) </syntaxhighlight></blockquote> The result should be: <blockquote><syntaxhighlight lang="lua" line start=15> :table[1] is 10. table[2] is 20. table[3] is 30. </syntaxhighlight></blockquote> Let's define a dictionary of Asian languages: <blockquote><syntaxhighlight lang="lua" line highlight=1,3,4 start=16 copy> local lstAsian = {ja = "日本語", ko = "한국어", zh = "中文"} print(lstNum[2]) print(lstAsian.ja) print(tableToString(lstAsian)) </syntaxhighlight></blockquote> The Debug console supports Unicode: <blockquote><syntaxhighlight lang="lua" line start=20> 20 日本語 :table.ja is 日本語. table.zh is 中文. table.ko is 한국어. </syntaxhighlight></blockquote> CAUTION: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order. Note: press upwards arrow ↑ for the historic in the Debug console. == Create a Lua Script with Tables == To create a Lua script with tables: # Go to the top of [[Module:Sandbox]] in the editor. # Replace the existing code pasting the following Lua code without clicking on <code>Publish changes</code>. <blockquote><syntaxhighlight lang="lua" line highlight=2,14,15,21 copy> local p = {} local function tableToString(tab) -- pretty stringify input table local key; local val; local sep; local res = {':'} -- left margin for key, val in pairs(tab) do -- Iterates each pair key val if #res % 5 == 0 then sep = '\n:' else sep = '. ' end -- separator if tonumber(key) then -- numerical index of table? res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index] else -- alpha key without white separator res[#res + 1] = 'table.' .. key .. ' is ' .. val .. sep -- dot alpha end end return table.concat(res) -- concatenation more efficient than string `..` operator end function p.sequence() return ';sequence\n' .. tableToString({10, 20, 30}) end -- numerical sequence function p.dictionary() -- returns a dictionary of language codes -> English names local lstLang = {de = "German", en = "English", es = "Spanish", fr = "French", it = "Italian", -- Europe ja = "Japanese", ko = "Korean", ru = "Russian", zh = "Chinese" -- Asia + Russia } return ';dictionary\n' .. tableToString(lstLang) end function p.translate() -- provides language codes -> localized native names local lstLang = {de = "Deutsch", en = "English", es = "Español", fr = "Français", it = "Italiano", -- Latin ja = "日本語", ko = "한국어", ru = "Русский", zh = "中文" -- Unicode } return ';translate\n' .. tableToString(lstLang) end return p </syntaxhighlight></blockquote> In <code>Debug console</code>, validate: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> print(p.sequence()) print(p.dictionary()) print(p.translate()) </syntaxhighlight></blockquote> WARNING: Iterator <syntaxhighlight lang="lua" inline>pairs(tab)</syntaxhighlight> may iterate in arbitrary order: <blockquote><syntaxhighlight lang="lua" line start=4> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Save the Module:Sandbox to call it from another Wiki page. # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1-3 copy> {{#invoke:Sandbox|sequence}} {{#invoke:Sandbox|dictionary}} {{#invoke:Sandbox|translate}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;sequence :table[1] is 10. table[2] is 20. table[3] is 30. ;dictionary :table.es is Spanish. table.ja is Japanese. table.fr is French. table.ru is Russian. table.de is German :table.ko is Korean. table.en is English. table.zh is Chinese. table.it is Italian. ;translate :table.es is Español. table.ja is 日本語. table.fr is Français. table.ru is Русский. table.de is Deutsch :table.ko is 한국어. table.en is English. table.zh is 中文. table.it is Italiano. </blockquote> == Understand Your Lua Script == === tableToString === To understand your Lua script <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function tableToString(tab)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>, which is the table to be converted to a string. # <syntaxhighlight lang="lua" inline>local</syntaxhighlight> and the following code defines the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight>, <syntaxhighlight lang="lua" inline>val</syntaxhighlight>, and <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. All are <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # The previous lesson is based on string <syntaxhighlight lang="lua" inline>result = ''</syntaxhighlight>.<br>Let's introduce the table <syntaxhighlight lang="lua" inline>local res = {':'} -- left margin</syntaxhighlight> with its counter <syntaxhighlight lang="lua" inline>#res -- length of the table</syntaxhighlight> # <syntaxhighlight lang="lua" inline>for key, value in pairs(tab) do</syntaxhighlight> creates a loop code block that will vary the value of the variables <syntaxhighlight lang="lua" inline>key</syntaxhighlight> and <syntaxhighlight lang="lua" inline>value</syntaxhighlight> for each key/value data pair in the table <syntaxhighlight lang="lua" inline>tab</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>if tonumber(key) then</syntaxhighlight> tests to see if the key can be converted to a number. If it can, the key is displayed as a number (without quotes). If it can't be converted, the key is displayed as a literal string (with quotes). # Compare the string concatenation: <syntaxhighlight lang="lua" inline>result = result .. ':table[' .. key .. '] is ' .. value .. '\n'</syntaxhighlight> or<br><syntaxhighlight lang="lua" inline>result = result .. ':table[\'' .. key .. '\'] is ' .. value .. '\n'</syntaxhighlight> adds the current key and value to the result.<br><syntaxhighlight lang="lua" inline>res[#res + 1] = 'table[' .. key .. '] is ' .. val .. sep -- [index]</syntaxhighlight> will be faster thanks to <syntaxhighlight lang="lua" inline>return table.concat(res)</syntaxhighlight>. #* The <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation is a common way to reference individual table keys to access their associated values. #* To display quotes inside a string, you must either switch between quote identifiers ("/'), or use the \ character to 'escape' the quote, which forces it to be part of the string rather than terminate the string. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the loop. # <syntaxhighlight lang="lua" inline>return table.concat(res) -- concatenation more efficient than string `..` operator</syntaxhighlight> returns the concatenation of each item in the table as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.sequence === To understand your Lua script <syntaxhighlight lang="lua" inline>sequence</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.sequence() return ';sequence\n' .. tableToString({10, 20, 30}) end -- numerical sequence</syntaxhighlight> could be detailed in the following steps. # <syntaxhighlight lang="lua" inline>local numbers = {10, 20, 30}</syntaxhighlight> defines a local table variable named <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight>, which is initialized with three values. #: When table values are specified without matching keys, Lua automatically adds numeric keys for the values as a sequence from 1 to N, the number of values added. This allows Lua tables to be used similar to how arrays are used in other programming languages. For example, the second value in the table could be referenced as <syntaxhighlight lang="lua" inline>numbers[2]</syntaxhighlight>, and the table values could be processed with a for loop. # <syntaxhighlight lang="lua" inline>local result</syntaxhighlight> defines the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight> and initializes it to <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>result = ';sequence\n'</syntaxhighlight> assigns a literal string value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. The semi colon is an operator that separates each new statement and the /n means new line. # <syntaxhighlight lang="lua" inline>result = result .. tableToString(numbers)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>numbers</syntaxhighlight> as the parameter and concatenates the returned value to the variable <syntaxhighlight lang="lua" inline>result</syntaxhighlight>. # <syntaxhighlight lang="lua" inline>return result</syntaxhighlight> returns the current value of <syntaxhighlight lang="lua" inline>result</syntaxhighlight> as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === p.dictionary or p.translate === To understand your Lua script <syntaxhighlight lang="lua" inline>dictionary</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.dictionary()</syntaxhighlight> declares a function named dictionary. # <syntaxhighlight lang="lua" inline>local lstLang = {de = "German", en = "English",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>ja = "日本語", ko = "한국어",</syntaxhighlight> ... <syntaxhighlight lang="lua" inline>, zh = "中文"} -- Unicode</syntaxhighlight> and the following code defines a local table variable named <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight>, which is initialized with 9 key/value pairs. #* When table values are specified with matching keys, Lua uses the specified keys instead of numeric key values. #* Lua tables with specified key values cannot be used as arrays using <syntaxhighlight lang="lua" inline>table[number]</syntaxhighlight> notation. #* Lua tables with specified key values can be used as associative arrays using <syntaxhighlight lang="lua" inline>table[key]</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang['ja']</syntaxhighlight>. Quotes around string literal keys are required with this notation format. #* Lua tables with specified key values can also be used as associative arrays using <syntaxhighlight lang="lua" inline>table.key</syntaxhighlight> notation, such as <syntaxhighlight lang="lua" inline>lstLang.ja</syntaxhighlight>. Quotes around string literal keys are not required with this notation format. # <syntaxhighlight lang="lua" inline>return ';dictionary\n' .. tableToString(lstLang)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>tableToString</syntaxhighlight> function, passing the table <syntaxhighlight lang="lua" inline>lstLang</syntaxhighlight> as the parameter and concatenates the returned value as the result of the function. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. It is important to note that the for loop to process table pairs will process all values in the table, but tables with specified key values may not be processed in the order they were created. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with tables. Continue on to the [[../Errors/]] lesson. == See Also == * [[Lua/Table Library | Lua Table Library]] * [[Wikipedia: Associative array]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] i60avdmph18wtq1meoi9n4326lo3gtx Lua/Errors 0 153546 2803401 2141601 2026-04-07T20:20:29Z Nimmzo 801528 /* lead section */ +syntaxhighlight +inline +line +highlight +copy +[[Template:cite web]] http → https 2803401 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a table to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-07}}</ref> This lesson will show you how to troubleshoot script errors and handle run-time errors in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Tables| Tables]] lesson. == Create a Lua Script with Errors and Error Handling == To create a Lua script with errors and error handling: # Navigate to [[Module:Sandbox]]. # Clear all existing code. #: It's a sandbox. Everyone is free to play in the sandbox. But if you find another user is actively editing the sandbox at the same time, you may also use Module:Sandbox/Username, where Username is your Wikiversity username. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=4,22,34 copy> local p = {} local function reciprocal1(value) return 1 / value end function p.test1(frame) local value = frame.args[1] return reciprocal1(value) end local function reciprocal2(value) if value == nil then error('value must exist') end if tonumber(value) == nil then error('value must be a number') end if tonumber(value) == 0 then error('value must not be 0') end return 1 / value end function p.test2(frame) local value = frame.args[1] return reciprocal2(value) end local function reciprocal3(value) assert(value, 'value must exist') assert(tonumber(value), 'value must be a number') assert(tonumber(value) ~= 0, 'value must not be zero') return 1 / value end function p.test3(frame) local value = frame.args[1] return reciprocal3(value) end function p.test4(frame) local value = frame.args[1] if pcall(function () result = reciprocal3(value) end) then return result else return 'Error: Value must exist, must be numeric, and not zero.' end end return p </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><pre> ;Reciprocal 1 :{{#invoke:Sandbox|test1}} :{{#invoke:Sandbox|test1|x}} :{{#invoke:Sandbox|test1|0}} :{{#invoke:Sandbox|test1|2}} ;Reciprocal 2 :{{#invoke:Sandbox|test2}} :{{#invoke:Sandbox|test2|x}} :{{#invoke:Sandbox|test2|0}} :{{#invoke:Sandbox|test2|2}} ;Reciprocal 3 :{{#invoke:Sandbox|test3}} :{{#invoke:Sandbox|test3|x}} :{{#invoke:Sandbox|test3|0}} :{{#invoke:Sandbox|test3|2}} ;Reciprocal 4 :{{#invoke:Sandbox|test4}} :{{#invoke:Sandbox|test4|x}} :{{#invoke:Sandbox|test4|0}} :{{#invoke:Sandbox|test4|2}} </pre></blockquote> The result should be similar to: <blockquote> ;Reciprocal 1 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :inf :0.5 ;Reciprocal 2 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :0.5 ;Reciprocal 3 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :0.5 ;Reciprocal 4 :Error: Value must exist, must be numeric, and not zero. :Error: Value must exist, must be numeric, and not zero. :Error: Value must exist, must be numeric, and not zero. :0.5 </blockquote> == Understand Your Lua Script == To understand your Lua script <code>reciprocal1</code> function: # <code>local function reciprocal1(value)</code> declares a local function named <code>reciprocal1</code> that accepts a single parameter <code>value</code>, which is the value whose reciprocal will be returned. # <code>return 1 / value</code> returns the reciprocal of value. # <code>end</code> ends the function. To understand your Lua script <code>test1</code> function: # <code>function p.test1(frame)</code> declares a function named <code>test1</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>return reciprocal1(value)</code> calls the <code>reciprocal1</code> function, passing the variable <code>value</code> and returns the result. # <code>end</code> ends the function. To understand your Lua script <code>reciprocal2</code> function: # <code>local function reciprocal2(value)</code> declares a local function named <code>reciprocal2</code> that accepts a single parameter <code>value</code>, which is the value whose reciprocal will be returned. # <code>if value == nil then</code> creates a conditional code block and tests to see if <code>value</code> is <code>nil</code>. If it is, an error is generated. # <code>error('value must exist')</code> generates an error with the given literal string as the error statement. Using <code>error()</code> allows the script writer to determine the text of the error message that is returned to the calling function or wiki page that invoked the function. #: When an error is generated, execution immediately returns to the calling function. Any additional code in the same function that comes after the error is not processed. # <code>if tonumber(value) == nil</code> creates a conditional code block and tests to see if <code>value</code> is numeric. If it is not, <code>tonumber</code> returns <code>nil</code> and an error is generated. # <code>if tonumber(value) == 0</code> creates a conditional code block and tests to see if <code>value</code> is <code>0</code>. If it is, an error is generated. # <code>return 1 / value</code> returns the reciprocal of value. # <code>end</code> ends the function. To understand your Lua script <code>test2</code> function: # <code>function p.test2(frame)</code> and the following code declares a function named <code>test2</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>return reciprocal2(value)</code> calls the <code>reciprocal2</code> function, passing the variable <code>value</code> and returns the result. # <code>end</code> ends the function. To understand your Lua script <code>reciprocal3</code> function: # <code>local function reciprocal3(value)</code> declares a local function named <code>reciprocal3</code> that accepts a single parameter <code>value</code>, which is the value whose reciprocal will be returned. # <code>assert(value, 'value must exist')</code> creates a self-contained conditional code block and tests to see if <code>value</code> is <code>nil</code> If it is, an error is generated. #: It is best practice to use <code>assert</code> to document and test any assumptions that are made regarding passed parameters. # <code>assert(tonumber(value), 'value must be a number')</code> creates a self-contained conditional code block and tests to see if <code>value</code> is numeric. If it is not, <code>tonumber</code> returns <code>nil</code> and an error is generated. # <code>assert(tonumber(value) ~= 0, 'value must not be zero')</code> creates a conditional code block and tests to see if <code>value</code> is <code>0</code>. If it is, an error is generated. # <code>return 1 / value</code> returns the reciprocal of value. # <code>end</code> ends the function. To understand your Lua script <code>test3</code> function: # <code>function p.test3(frame)</code> and the following code declares a function named <code>test3</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>return reciprocal3(value)</code> calls the <code>reciprocal3</code> function, passing the variable <code>value</code> and returns the result. # <code>end</code> ends the function. To understand your Lua script <code>test4</code> function: # <code>function p.test4(frame)</code> and the following code declares a function named <code>test4</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>if pcall(function () result = reciprocal3(value) end) then</code> creates a conditional code block that calls the <code>reciprocal3</code> function, passing <code>value</code> and storing the result. If no error occurs, <code>result</code> is returned. If an error occurs, a literal string is returned instead. # <code>pcall()</code> uses the pcall (protected call) function to call a function and catch any errors that occur. # <code>function () ... end</code> creates a self-contained anonymous function that executes the code in ... as a function call. #:The anonymous function is necessary to save the result of the <code>reciprocal3</code> function while also using <code>pcall()</code> to catch any errors that occur. # <code>end</code> ends the function. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with error handling. Continue on to the [[../Math Library/]] lesson or return to the main [[Lua]] page to learn about other Lua code libraries. == See Also == * [[Wikipedia: Exception handling]] * [[Wikipedia: Assertion (software development)]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] erjp1work4hafyt5m4w610johwg6og4 2803403 2803401 2026-04-07T20:30:15Z Nimmzo 801528 /* Test Your Lua Script */ pre → syntaxhighlight +line +highlight +copy 2803403 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a table to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-07}}</ref> This lesson will show you how to troubleshoot script errors and handle run-time errors in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Tables| Tables]] lesson. == Create a Lua Script with Errors and Error Handling == To create a Lua script with errors and error handling: # Navigate to [[Module:Sandbox]]. # Clear all existing code. #: It's a sandbox. Everyone is free to play in the sandbox. But if you find another user is actively editing the sandbox at the same time, you may also use Module:Sandbox/Username, where Username is your Wikiversity username. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=4,22,34 copy> local p = {} local function reciprocal1(value) return 1 / value end function p.test1(frame) local value = frame.args[1] return reciprocal1(value) end local function reciprocal2(value) if value == nil then error('value must exist') end if tonumber(value) == nil then error('value must be a number') end if tonumber(value) == 0 then error('value must not be 0') end return 1 / value end function p.test2(frame) local value = frame.args[1] return reciprocal2(value) end local function reciprocal3(value) assert(value, 'value must exist') assert(tonumber(value), 'value must be a number') assert(tonumber(value) ~= 0, 'value must not be zero') return 1 / value end function p.test3(frame) local value = frame.args[1] return reciprocal3(value) end function p.test4(frame) local value = frame.args[1] if pcall(function () result = reciprocal3(value) end) then return result else return 'Error: Value must exist, must be numeric, and not zero.' end end return p </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1,6,11,16 copy> ;Reciprocal 1 :{{#invoke:Sandbox|test1}} :{{#invoke:Sandbox|test1|x}} :{{#invoke:Sandbox|test1|0}} :{{#invoke:Sandbox|test1|2}} ;Reciprocal 2 :{{#invoke:Sandbox|test2}} :{{#invoke:Sandbox|test2|x}} :{{#invoke:Sandbox|test2|0}} :{{#invoke:Sandbox|test2|2}} ;Reciprocal 3 :{{#invoke:Sandbox|test3}} :{{#invoke:Sandbox|test3|x}} :{{#invoke:Sandbox|test3|0}} :{{#invoke:Sandbox|test3|2}} ;Reciprocal 4 :{{#invoke:Sandbox|test4}} :{{#invoke:Sandbox|test4|x}} :{{#invoke:Sandbox|test4|0}} :{{#invoke:Sandbox|test4|2}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;Reciprocal 1 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :inf :0.5 ;Reciprocal 2 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :0.5 ;Reciprocal 3 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :0.5 ;Reciprocal 4 :Error: Value must exist, must be numeric, and not zero. :Error: Value must exist, must be numeric, and not zero. :Error: Value must exist, must be numeric, and not zero. :0.5 </blockquote> == Understand Your Lua Script == To understand your Lua script <code>reciprocal1</code> function: # <code>local function reciprocal1(value)</code> declares a local function named <code>reciprocal1</code> that accepts a single parameter <code>value</code>, which is the value whose reciprocal will be returned. # <code>return 1 / value</code> returns the reciprocal of value. # <code>end</code> ends the function. To understand your Lua script <code>test1</code> function: # <code>function p.test1(frame)</code> declares a function named <code>test1</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>return reciprocal1(value)</code> calls the <code>reciprocal1</code> function, passing the variable <code>value</code> and returns the result. # <code>end</code> ends the function. To understand your Lua script <code>reciprocal2</code> function: # <code>local function reciprocal2(value)</code> declares a local function named <code>reciprocal2</code> that accepts a single parameter <code>value</code>, which is the value whose reciprocal will be returned. # <code>if value == nil then</code> creates a conditional code block and tests to see if <code>value</code> is <code>nil</code>. If it is, an error is generated. # <code>error('value must exist')</code> generates an error with the given literal string as the error statement. Using <code>error()</code> allows the script writer to determine the text of the error message that is returned to the calling function or wiki page that invoked the function. #: When an error is generated, execution immediately returns to the calling function. Any additional code in the same function that comes after the error is not processed. # <code>if tonumber(value) == nil</code> creates a conditional code block and tests to see if <code>value</code> is numeric. If it is not, <code>tonumber</code> returns <code>nil</code> and an error is generated. # <code>if tonumber(value) == 0</code> creates a conditional code block and tests to see if <code>value</code> is <code>0</code>. If it is, an error is generated. # <code>return 1 / value</code> returns the reciprocal of value. # <code>end</code> ends the function. To understand your Lua script <code>test2</code> function: # <code>function p.test2(frame)</code> and the following code declares a function named <code>test2</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>return reciprocal2(value)</code> calls the <code>reciprocal2</code> function, passing the variable <code>value</code> and returns the result. # <code>end</code> ends the function. To understand your Lua script <code>reciprocal3</code> function: # <code>local function reciprocal3(value)</code> declares a local function named <code>reciprocal3</code> that accepts a single parameter <code>value</code>, which is the value whose reciprocal will be returned. # <code>assert(value, 'value must exist')</code> creates a self-contained conditional code block and tests to see if <code>value</code> is <code>nil</code> If it is, an error is generated. #: It is best practice to use <code>assert</code> to document and test any assumptions that are made regarding passed parameters. # <code>assert(tonumber(value), 'value must be a number')</code> creates a self-contained conditional code block and tests to see if <code>value</code> is numeric. If it is not, <code>tonumber</code> returns <code>nil</code> and an error is generated. # <code>assert(tonumber(value) ~= 0, 'value must not be zero')</code> creates a conditional code block and tests to see if <code>value</code> is <code>0</code>. If it is, an error is generated. # <code>return 1 / value</code> returns the reciprocal of value. # <code>end</code> ends the function. To understand your Lua script <code>test3</code> function: # <code>function p.test3(frame)</code> and the following code declares a function named <code>test3</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>return reciprocal3(value)</code> calls the <code>reciprocal3</code> function, passing the variable <code>value</code> and returns the result. # <code>end</code> ends the function. To understand your Lua script <code>test4</code> function: # <code>function p.test4(frame)</code> and the following code declares a function named <code>test4</code> that accepts a single parameter <code>frame</code>, which is the object used to access parameters passed from #invoke. # <code>local value = frame.args[1]</code> defines a local variable named <code>value</code> and assigns the value of the first frame argument (parameter) passed with #invoke. # <code>if pcall(function () result = reciprocal3(value) end) then</code> creates a conditional code block that calls the <code>reciprocal3</code> function, passing <code>value</code> and storing the result. If no error occurs, <code>result</code> is returned. If an error occurs, a literal string is returned instead. # <code>pcall()</code> uses the pcall (protected call) function to call a function and catch any errors that occur. # <code>function () ... end</code> creates a self-contained anonymous function that executes the code in ... as a function call. #:The anonymous function is necessary to save the result of the <code>reciprocal3</code> function while also using <code>pcall()</code> to catch any errors that occur. # <code>end</code> ends the function. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with error handling. Continue on to the [[../Math Library/]] lesson or return to the main [[Lua]] page to learn about other Lua code libraries. == See Also == * [[Wikipedia: Exception handling]] * [[Wikipedia: Assertion (software development)]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] 48nhkiva2hq0s9l6cmpo7lua3ffbg4i 2803409 2803403 2026-04-07T20:40:06Z Nimmzo 801528 /* Understand Your Lua Script */ +syntaxhighlight +inline 2803409 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Lua modules based on the Scribunto/Lua extension are stored in resource pages using the <syntaxhighlight lang="lua" inline>Module:</syntaxhighlight> namespace. Each module uses a table to hold functions and variables, and that containing table is returned at the end of the module code.<ref>{{cite web |date=2026 |title=Extension:Scribunto/Lua reference manual |url=https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table |website=Mediawiki |access-date=2026-04-07}}</ref> This lesson will show you how to troubleshoot script errors and handle run-time errors in your scripts. __TOC__ == Prerequisites == This lesson assumes you have already completed the [[Lua/Tables| Tables]] lesson. == Create a Lua Script with Errors and Error Handling == To create a Lua script with errors and error handling: # Navigate to [[Module:Sandbox]]. # Clear all existing code. #: It's a sandbox. Everyone is free to play in the sandbox. But if you find another user is actively editing the sandbox at the same time, you may also use Module:Sandbox/Username, where Username is your Wikiversity username. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=4,22,34 copy> local p = {} local function reciprocal1(value) return 1 / value end function p.test1(frame) local value = frame.args[1] return reciprocal1(value) end local function reciprocal2(value) if value == nil then error('value must exist') end if tonumber(value) == nil then error('value must be a number') end if tonumber(value) == 0 then error('value must not be 0') end return 1 / value end function p.test2(frame) local value = frame.args[1] return reciprocal2(value) end local function reciprocal3(value) assert(value, 'value must exist') assert(tonumber(value), 'value must be a number') assert(tonumber(value) ~= 0, 'value must not be zero') return 1 / value end function p.test3(frame) local value = frame.args[1] return reciprocal3(value) end function p.test4(frame) local value = frame.args[1] if pcall(function () result = reciprocal3(value) end) then return result else return 'Error: Value must exist, must be numeric, and not zero.' end end return p </syntaxhighlight></blockquote> == Test Your Lua Script == To test your Lua script: # Navigate to either the [[Module_talk:Sandbox]] page, the [[Wikiversity:Sandbox]] page, or your own user or sandbox page. # Add the following code and save the page: <blockquote><syntaxhighlight lang="lua" line highlight=1,6,11,16 copy> ;Reciprocal 1 :{{#invoke:Sandbox|test1}} :{{#invoke:Sandbox|test1|x}} :{{#invoke:Sandbox|test1|0}} :{{#invoke:Sandbox|test1|2}} ;Reciprocal 2 :{{#invoke:Sandbox|test2}} :{{#invoke:Sandbox|test2|x}} :{{#invoke:Sandbox|test2|0}} :{{#invoke:Sandbox|test2|2}} ;Reciprocal 3 :{{#invoke:Sandbox|test3}} :{{#invoke:Sandbox|test3|x}} :{{#invoke:Sandbox|test3|0}} :{{#invoke:Sandbox|test3|2}} ;Reciprocal 4 :{{#invoke:Sandbox|test4}} :{{#invoke:Sandbox|test4|x}} :{{#invoke:Sandbox|test4|0}} :{{#invoke:Sandbox|test4|2}} </syntaxhighlight></blockquote> The result should be similar to: <blockquote> ;Reciprocal 1 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :inf :0.5 ;Reciprocal 2 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :0.5 ;Reciprocal 3 :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :{{fontcolor|#CC0000|'''Script error'''}} :0.5 ;Reciprocal 4 :Error: Value must exist, must be numeric, and not zero. :Error: Value must exist, must be numeric, and not zero. :Error: Value must exist, must be numeric, and not zero. :0.5 </blockquote> == Understand Your Lua Script == === reciprocal1 === To understand your Lua script <syntaxhighlight lang="lua" inline>reciprocal1</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function reciprocal1(value)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>reciprocal1</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>value</syntaxhighlight>, which is the value whose reciprocal will be returned. # <syntaxhighlight lang="lua" inline>return 1 / value</syntaxhighlight> returns the reciprocal of value. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === test1 === To understand your Lua script <syntaxhighlight lang="lua" inline>test1</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.test1(frame)</syntaxhighlight> declares a function named <syntaxhighlight lang="lua" inline>test1</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>frame</syntaxhighlight>, which is the object used to access parameters passed from #invoke. # <syntaxhighlight lang="lua" inline>local value = frame.args[1]</syntaxhighlight> defines a local variable named <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and assigns the value of the first frame argument (parameter) passed with #invoke. # <syntaxhighlight lang="lua" inline>return reciprocal1(value)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>reciprocal1</syntaxhighlight> function, passing the variable <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and returns the result. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === reciprocal2 === To understand your Lua script <syntaxhighlight lang="lua" inline>reciprocal2</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function reciprocal2(value)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>reciprocal2</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>value</syntaxhighlight>, which is the value whose reciprocal will be returned. # <syntaxhighlight lang="lua" inline>if value == nil then</syntaxhighlight> creates a conditional code block and tests to see if <syntaxhighlight lang="lua" inline>value</syntaxhighlight> is <syntaxhighlight lang="lua" inline>nil</syntaxhighlight>. If it is, an error is generated. # <syntaxhighlight lang="lua" inline>error('value must exist')</syntaxhighlight> generates an error with the given literal string as the error statement. Using <syntaxhighlight lang="lua" inline>error()</syntaxhighlight> allows the script writer to determine the text of the error message that is returned to the calling function or wiki page that invoked the function. #: When an error is generated, execution immediately returns to the calling function. Any additional code in the same function that comes after the error is not processed. # <syntaxhighlight lang="lua" inline>if tonumber(value) == nil</syntaxhighlight> creates a conditional code block and tests to see if <syntaxhighlight lang="lua" inline>value</syntaxhighlight> is numeric. If it is not, <syntaxhighlight lang="lua" inline>tonumber</syntaxhighlight> returns <syntaxhighlight lang="lua" inline>nil</syntaxhighlight> and an error is generated. # <syntaxhighlight lang="lua" inline>if tonumber(value) == 0</syntaxhighlight> creates a conditional code block and tests to see if <syntaxhighlight lang="lua" inline>value</syntaxhighlight> is <syntaxhighlight lang="lua" inline>0</syntaxhighlight>. If it is, an error is generated. # <syntaxhighlight lang="lua" inline>return 1 / value</syntaxhighlight> returns the reciprocal of value. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === test2 === To understand your Lua script <syntaxhighlight lang="lua" inline>test2</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.test2(frame)</syntaxhighlight> and the following code declares a function named <syntaxhighlight lang="lua" inline>test2</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>frame</syntaxhighlight>, which is the object used to access parameters passed from #invoke. # <syntaxhighlight lang="lua" inline>local value = frame.args[1]</syntaxhighlight> defines a local variable named <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and assigns the value of the first frame argument (parameter) passed with #invoke. # <syntaxhighlight lang="lua" inline>return reciprocal2(value)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>reciprocal2</syntaxhighlight> function, passing the variable <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and returns the result. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === reciprocal3 === To understand your Lua script <syntaxhighlight lang="lua" inline>reciprocal3</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>local function reciprocal3(value)</syntaxhighlight> declares a local function named <syntaxhighlight lang="lua" inline>reciprocal3</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>value</syntaxhighlight>, which is the value whose reciprocal will be returned. # <syntaxhighlight lang="lua" inline>assert(value, 'value must exist')</syntaxhighlight> creates a self-contained conditional code block and tests to see if <syntaxhighlight lang="lua" inline>value</syntaxhighlight> is <syntaxhighlight lang="lua" inline>nil</syntaxhighlight> If it is, an error is generated. #: It is best practice to use <syntaxhighlight lang="lua" inline>assert</syntaxhighlight> to document and test any assumptions that are made regarding passed parameters. # <syntaxhighlight lang="lua" inline>assert(tonumber(value), 'value must be a number')</syntaxhighlight> creates a self-contained conditional code block and tests to see if <syntaxhighlight lang="lua" inline>value</syntaxhighlight> is numeric. If it is not, <syntaxhighlight lang="lua" inline>tonumber</syntaxhighlight> returns <syntaxhighlight lang="lua" inline>nil</syntaxhighlight> and an error is generated. # <syntaxhighlight lang="lua" inline>assert(tonumber(value) ~= 0, 'value must not be zero')</syntaxhighlight> creates a conditional code block and tests to see if <syntaxhighlight lang="lua" inline>value</syntaxhighlight> is <syntaxhighlight lang="lua" inline>0</syntaxhighlight>. If it is, an error is generated. # <syntaxhighlight lang="lua" inline>return 1 / value</syntaxhighlight> returns the reciprocal of value. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === test3 === To understand your Lua script <syntaxhighlight lang="lua" inline>test3</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.test3(frame)</syntaxhighlight> and the following code declares a function named <syntaxhighlight lang="lua" inline>test3</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>frame</syntaxhighlight>, which is the object used to access parameters passed from #invoke. # <syntaxhighlight lang="lua" inline>local value = frame.args[1]</syntaxhighlight> defines a local variable named <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and assigns the value of the first frame argument (parameter) passed with #invoke. # <syntaxhighlight lang="lua" inline>return reciprocal3(value)</syntaxhighlight> calls the <syntaxhighlight lang="lua" inline>reciprocal3</syntaxhighlight> function, passing the variable <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and returns the result. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. === test4 === To understand your Lua script <syntaxhighlight lang="lua" inline>test4</syntaxhighlight> function: # <syntaxhighlight lang="lua" inline>function p.test4(frame)</syntaxhighlight> and the following code declares a function named <syntaxhighlight lang="lua" inline>test4</syntaxhighlight> that accepts a single parameter <syntaxhighlight lang="lua" inline>frame</syntaxhighlight>, which is the object used to access parameters passed from #invoke. # <syntaxhighlight lang="lua" inline>local value = frame.args[1]</syntaxhighlight> defines a local variable named <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and assigns the value of the first frame argument (parameter) passed with #invoke. # <syntaxhighlight lang="lua" inline>if pcall(function () result = reciprocal3(value) end) then</syntaxhighlight> creates a conditional code block that calls the <syntaxhighlight lang="lua" inline>reciprocal3</syntaxhighlight> function, passing <syntaxhighlight lang="lua" inline>value</syntaxhighlight> and storing the result. If no error occurs, <syntaxhighlight lang="lua" inline>result</syntaxhighlight> is returned. If an error occurs, a literal string is returned instead. # <syntaxhighlight lang="lua" inline>pcall()</syntaxhighlight> uses the pcall (protected call) function to call a function and catch any errors that occur. # <syntaxhighlight lang="lua" inline>function () ... end</syntaxhighlight> creates a self-contained anonymous function that executes the code in ... as a function call. #:The anonymous function is necessary to save the result of the <syntaxhighlight lang="lua" inline>reciprocal3</syntaxhighlight> function while also using <syntaxhighlight lang="lua" inline>pcall()</syntaxhighlight> to catch any errors that occur. # <syntaxhighlight lang="lua" inline>end</syntaxhighlight> ends the function. == Conclusion == Congratulations! You've now created, tested, and understood a Lua script with error handling. Continue on to the [[../Math Library/]] lesson or return to the main [[Lua]] page to learn about other Lua code libraries. == See Also == * [[Wikipedia: Exception handling]] * [[Wikipedia: Assertion (software development)]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category: Lessons]] [[Category: Completed resources]] ogrt57kt388m7cfugb56eycmldyparv Alphabet/Spanish alphabet 0 154046 2803555 2796318 2026-04-08T11:01:06Z ~2026-21712-42 3064537 https://heartumental.org/ 2803555 wikitext text/x-wiki The Spanish alphabet has 27 letters. Synonyms: [https://heartumental.org/ alfabeto] (from Ancient Greek''ἄλφα/alfa/alpha'', ''beta''...) and abecedario (from Spanish sounding: a-be-ce-da-rio). [[file:Es-alphabet-Spain.oga]] ---- Notes: #Only vocals may contain an accent (''' ' '''). Only the letter "u" may show an accent or dieresis ('''ü'''). #The letter "'''H'''" or "'''h'''" is silent, unless it is preceded by the letter "'''C'''" or "'''c'''" to form "'''''Ch'''''" (or by the letter '''S''' to form '''''Sh'''''). #The combination of the letters '''P''' or '''p''' and '''h''' used in English is not used in Spanish, particularly to produce the '''''f''''' sound; so words like ''alpha'' (in English) are written as ''alfa'' in Spanish. #The letter "'''Q'''" or "'''q'''" is basically accompanied by the letter "'''u'''", to be followed by a vocal, such as "'''e'''" or "'''i'''" to form the syllable ''que'' and ''qui''. #The letter "'''W'''" or "'''w'''", as the combination "'''Sh'''" or "'''sh'''" (and in some instances the letter "'''x'''" or "'''x'''") present flexibility to accommodate some neologisms, anglicisms, and other terms and names not necessarily original in Spanish. For instance: '''''W'''a'''sh'''ington''. #The letter "'''X'''" or "'''x'''", encompasses different sounds in pronunciation depending on the use of the word and the position of the letter in the word. ---- {|class="wikitable" |- !# !Letter (Uppercase) !Letter (Lowercase) !Pronunciation (Name of the letter) !Sound (Name of the letter) |- |1 |A |a (á) |a |[[file:letter a es es.flac]] |- |2 |B |b |be |[[file:letter b es es.flac]] |- |3 |C |c |ce |[[file:letter c es es.flac]] |- |4 |D |d |de |[[file:letter d es es.flac]] |- |5 |E |e (é) |e |[[file:letter e es es.flac]] |- |6 |F |f |efe |[[file:letter f es es.flac]] |- |7 |G |g |ge |[[file:letter g es es.flac]] |- |8 |H |h |atcheh |[[file:letter h es es.flac]] |- |9 |I |i (í) |i |[[file:letter i es es.flac]] |- |10 |J |j |jota |[[file:letter j es es.flac]] |- |11 |K |k |ka |[[file:letter k es es.flac]] |- |12 |L |l |ele |[[file:letter l es es.flac]] |- |13 |M |m |eme |[[file:letter m es es.flac]] |- |14 |N |n |ene |[[file:letter n es es.flac]] |- |15 |Ñ |ñ |eñe |[[File:Letter ñ es es.flac]] |- |16 |O |o (ó) |o |[[file:letter o es es.flac]] |- |17 |P |p |pe |[[file:letter p es es.flac]] |- |18 |Q |q |coo |[[file:letter q es es.flac]] |- |19 |R |r |ere |[[file:letter r es es.flac]] |- |20 |S |s |ese |[[file:letter s es es.flac]] |- |21 |T |t |te |[[file:letter t es es.flac]] |- |22 |U |u (ú, ü) |u |[[file:letter u es es.flac]] |- |23 |V |v |uve |[[file:letter v es es.flac]] |- |24 |W |w |doble uve |[[file:letter w es es.flac]] |- |25 |X |x |equis |[[file:letter x es es.flac]] |- |26 |Y |y |ye, i griega |[[File:Letter y recommended es es.flac]][[file:letter y es es.flac]] |- |27 |Z |z |zeta |[[file:letter z es es.flac]] |- |} * [[Wikipedia: Spanish alphabet]] * [[Portal: Spanish]] [[Category:Alphabets]] [[Category:Spanish]] [[Category:Spanish One]] apy7l8ibh13d81zbkrbg7enc8mu78tm Complex analysis in plain view 0 171005 2803334 2802871 2026-04-07T14:14:23Z Young1lim 21186 /* Geometric Series Examples */ 2803334 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260406.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] axauhw8o8kg9ey3dts51dvwuy4182u8 Universal Bibliography 0 171301 2803353 2802957 2026-04-07T16:12:15Z James500 297601 /* Cinema */Add 2803353 wikitext text/x-wiki {{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}} {{Bibliography}} {{research}} If this resource is ever completed, it will be a universal bibliography.<ref>See [[w:Bibliography]].</ref> Until then, it will be an approximation of a universal bibliography. This bibliography is arranged as an index of topics. ==Index== *[[Universal Bibliography/Bibliography|Bibliography]] *[[Universal Bibliography/Libraries|Libraries]] *[[Universal Bibliography/Literature|Literature]] *[[Universal Bibliography/SF|SF]] *[[Universal Bibliography/Music|Music]] *[[Universal Bibliography/Publishers and imprints|Publishers and imprints]] *[[Universal Bibliography/Printing|Printing]] *[[Universal Bibliography/Printers|Printers]] *[[Universal Bibliography/Microform|Microform]] *[[Universal Bibliography/Periodicals|Periodicals]] *[[Universal Bibliography/Reference|Reference]] *[[Universal Bibliography/Gazetteers|Gazetteers]] *[[Universal Bibliography/Humanities|Humanities]] *[[Universal Bibliography/Law|Law]] *[[Universal Bibliography/History|History]] *[[Universal Bibliography/Archaeology|Archaeology]] *[[Universal Bibliography/Geography|Geography]] *[[Universal Bibliography/Countries|Countries]] *[[Universal Bibliography/Architecture|Architecture]] *[[Universal Bibliography/Mathematics|Mathematics]] *[[Universal Bibliography/Computers|Computers]] *[[Universal Bibliography/Kites|Kites]] *[[Universal Bibliography/Nostalgia|Nostalgia]] *[[Universal Bibliography/Children's non-fiction|Children's non-fiction]] ===About=== *[[Universal Bibliography/About|About]] ==Online libraries== Swedish: *[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank) *[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg) ==Biographical dictionaries etc== See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]] *Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur, "Biographical Dictionaries in the Digital Era". Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&pg=PA83#v=onepage&q&f=false Page 83] et seq. Bibliographies, indexes, etc: *Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ] *Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ] *Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ] *Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ] *Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ] *Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ] *Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ] Particular works: *Oxford Dictionary of National Biography; Dictionary of National Biography *Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&btnG=Search+Books&bksoutput=html_text&tbm=bks&tbo=1 editions:uzt3-qMuFcMC]) *A & C Black's Who's Who *Who Was Who *The Academic Who's Who. A & C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&pg=PA208&output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC *Dictionary of Edwardian Biography (Pike's New Century Series) Works that comprise largely of biographies: *The Penguin Companion to Literature Theatres *A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers & Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15] ==Dictionaries of dates== [https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org] *Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ] *Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&tbm=bks&q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC] *Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880 *[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ] *Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440] *Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930. *[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]] *Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates. *Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will] *Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA93&output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA351&output=html_text] *Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May) *Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&id=k3dJAAAAYAAJ&output=html] *Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880] Australia *John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog] *John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford & Co. Brisbane. 1895. America *Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921. ==Commodity dictionaries== *Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&pg=PR22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&pg=RA15-PA7#v=onepage&q&f=false] *Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, "Commodity Dictionary", Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&pg=PA182#v=onepage&q&f=false p 182] & 183 *Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 & 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ] *Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC] *Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&pg=PA78#v=onepage&q&f=false] ==Encyclopedias== See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]] *Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990. *The Dorling Kindersley Illustrated Family Encyclopedia ==Almanacs== See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]]. *Year Book and Almanac of Newfoundland. **For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft] *Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C] ==Censuses== *Census of New Zealand and Labrador **1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog] **1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng] **1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng] ==Pilot guides== *[[w:United States Coast Pilot|United States Coast Pilot]] *American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&pg=PR1#v=onepage&q&f=false] *Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ] *Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ] *Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&pg=PR1#v=onepage&q&f=false] *Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&pg=PP5#v=onepage&q&f=false] ==Books of facts== *The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC *Chambers Book of Facts *Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008 *Handy Book of Facts: Things Everyone Should Know. C.S. Hammond & Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ] ==Series of books== See [[w:Category:Series of books]] and [[w:Category:Monographic series]] *George M Sinkankas, "Series" in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&pg=PA250#v=onepage&q&f=false 250] to 273. *"Publishing in Series, 1896-1916" in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&pg=PA539#v=onepage&q&f=false Page 539] et seq. *Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 2]. *Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ] *Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Khan. "Monographs in series". The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA208#v=onepage&q&f=false 207] to 209. *Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&pg=PA6#v=onepage&q&f=false 6] and 7. *Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&pg=PA11#v=onepage&q&f=false] **Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&pg=PA1190#v=onepage&q&f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&pg=PA53#v=onepage&q&f=false] ***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ] **Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ] **Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ] **Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ] *Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ] **2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ] **3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA63#v=onepage&q&f=false] *Ocran, Emmanuel Benjamin. Scientific & Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&pg=PA949#v=onepage&q&f=false] *Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ] *Young People's Literature in Series *Catalog of Reprints in Series. (sometimes called "Catalogue of Reprints in Series"). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA73#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ] *Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&pg=PA656#v=onepage&q&f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&pg=PA367#v=onepage&q&f=false] ==Series of classics== *Penguin Classics (Penguin Modern Classics, Penguin English Library) *Oxford World Classics *Everyman's Library *Wordsworth Classics *Macmillan Collectors Library *Bantam Classics *Minster Classics *The Literary Heritage Collection (Heron Books, London. William Collins Sons & Co, Glasgow) *Chandos Classics *Temple Classics *Longmans Heritage of Literature Series Russian *Greatest Masterpieces of Russian Literature (Heron Books, London) SF *Corgi SF Collectors Library Children's and shorter classics etc *Shorter Classics. Ginn and Company. *Ladybird Children's Classics. *Mini Classics. Parragon Books. *Bonny Books. Peter Haddock Ltd. *A series published by Dean & Son Ltd ==Non-fiction general series== *[[w:Oxford Companions|Oxford Companions]] *[[w:Cambridge Companions|Cambridge Companions]] *Princeton Companions *Blackwell Companions. Wiley Blackwell Companions *Routledge Companions. Routledge Research Companions *Ashgate Companions. Ashgate Research Companions *Brill's Companions *Facts on File Companions *Guides to Information Sources. Bowker-Saur *Butterworths Guides to Information Sources. *Columbia Guides *Blackwell Guides *Edinburgh Critical Guides *Collins Reference Dictionaries *New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]]) *Collins Gem (see [[w:List of Collins GEM books]]) *Concise Encyclopedias. Collins. *Time Life Books (see [[w:Time Life#Book series]]) *[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. *[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton. *Made Simple Books. W H Allen. *Palgrave Master Series *Harrap's Mini Series *Shire Albums. Shire Publications. *Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd. *The Wonderful World Books. Macdonald and Company *Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners. *Hamlyn Pocket Guides *Oxford Monograph Series *Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]] *Helpmate Handbooks. Willow Books University *University Paperbacks. Meuthen & Co *World Student Series. Addison Wesley *Unibooks. Hodder and Stoughton *International Student Editions. Van Nostrand Reinhold *Hutchinson University Library Imprints *Pelican Books Pictorials *Salmon Cameracolour series *Pitkin Pictorials United Kingdom *Aspects of Britain. HMSO. Places *The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]] *G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]] Art *Movements in World Art. Meuthen. *Movements in Modern Art. Meuthen. *How to Draw and Paint. New Burlington. Film *BFI Companions Popular science *Contemporary Science Paperbacks. Oliver and Boyd. *Pan Piper Science Series Science and mathematics *Simon and Schuster Tech Outlines *Schaum's Outline Series Military *Illustrated Military Guides. Illustrated Guides. "An Illustrated Guide to ...". Salamander Books. *Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press. *Osprey Men-at-Arms *Jane's Pocket Books Communication *The Library of Communication Techniques. Focal Press. *John Fiske (ed). Studies in Culture and Communication. Routledge. *The Media. Wayland. Cookery *ABC series. Peter Pauper Press. Gardening *Pan Piper Small Gardens Series. Mythology *Series on mythology published by Southwater (imprint of Anness) ==History and Geography== See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]]. *Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&pg=PP1#v=onepage&q&f=false] *Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&pg=PP1#v=onepage&q&f=false] General series *Cambridge Studies in Historical Geography Atlases *The Times Atlas of World History *Philip's Atlas of World History History of geography: *Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ] ==Chronology== See also [[Universal Bibliography/History#Millennia, centuries and decades]] General *Chronology of World History. **Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994. **Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994. **Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994. **Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon & Schuster. Reissued 1994. Centuries *Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ] *Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ] *Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC] *20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ] *Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ] *Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&pg=PP7#v=onepage&q&f=false] Decades *Series: **Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ] Years *Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&pg=PA113#v=onepage&q&f=false Catalogue]. *The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1882 to 1883]. *James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&pg=PP7#v=onepage&q&f=false] *[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&pg=PR1#v=onepage&q&f=false 1821]. *Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&pg=PR3#v=onepage&q&f=false] Ancient *Bickerman. Chronology of the Ancient World. 1968. *Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993. ==Anniversaries== *Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ] *On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&pg=PA1#v=onepage&q&f=false] ==Egyptology== *Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] ==Battlefields== *[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series] *"The Times Guide to Battlefields of Britain". Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart. ==Armed forces== Periodicals: *[[w:NATO Review|NATO Review]] Military *The Journal of Military History *Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC] *The Monthly Army List [Google editions:I0t2L4ElznEC] *The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC] *Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC] *The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&q=editions:dEauCcI7kssC&biw=534&bih=736&dpr=1.5#sbfbu=1] *The Royal Tank Corps Journal *The Tank [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:Dv-RbpoM7acC&biw=534&bih=736&dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment *The Cavalry Journal [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:cVQlfkRl6KUC&biw=534&bih=688&dpr=1.5#sbfbu=1] *The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&q=editions:liFy4uc0ggYC&biw=534&bih=736&dpr=1.5] *Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC] *The Royal Engineers Journal [https://www.google.com/search?tbm=bks&q=editions:8XobinXLbD0C&biw=534&bih=736&dpr=1.5] *Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C] *Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC] United States *Military Review *The Coast Artillery Journal [Google editions:nMCogSJ_rlkC] *Infantry Journal [Google editions:ULqoLmbUR5cC] *The Reserve Officer [Google editions:JQDRDrnD1QQC] Naval *[[w:Navy News|Navy News]] ==Armour== Armoured warfare; tank warfare *Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ] *Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ] *Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ] *Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ] *Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ] *Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&pg=PP1#v=onepage&q&f=false] *Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ] Atlas *Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ] Armored forces *Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ] ==Mesoamerica== *James. Aztecs & Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: "Books Received" (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and "Overviews for the general reader" (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252. *Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ] ==Accounting== See [[s:Category:Accounting]] Periodicals *[[s:The Accountant|The Accountant]] (1874 onwards) *Accountant's Magazine (1897 onwards) Aberdeen ==Arts== *Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997. ==Biography== *Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&pg=PP1#v=onepage&q&f=false] *Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ] *Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ] Yearbooks *Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC] *Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ] ==Information technology== *Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&pg=PR1#v=onepage&q&f=false] ==Economics== General series: *Dryden Press Series in Economics *Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational *Nuffield Economics & Business. Nuffield Foundation. Longman. Other: *Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984. *Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991. *Anderton, Alain. Economics. Causeway Press. 1991. *Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983. *Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991. *Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986. *Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989. *Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985.  *Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981. *National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990. *Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992. ==Games== Chess *Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987. *Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980. *Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977. *Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. General series *Batsford Chess Books **Discovering Chess Series. B T Batsford. Periodicals See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]] *British Chess Magazine Wargames *Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward. **Laurie Taylor. "Attila the Hun invades Tyne Tees". TV Times. 1978. pp 28 & 29. **Terry Wise. "Battleground". Battle for Wargamers. June 1978. pp 261 & 262. *[[w:Game of War|Game of War]]. Channel 4. 1997. [Television]. ==Cricket== See [[w:Bibliography of cricket]] *Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ]. **Peter Arnold. The Illustrated Encyclopedia of World Cricket. *Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ] Scores and biographies *Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&pg=PR3#v=onepage&q&f=false] **See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]] Periodicals *[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Australia *Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463] *The Oxford Companion to Australian Cricket India *The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies] Annuals *[[w:Indian Cricket (annual)|Indian Cricket]]. [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966]. ===Cricketers=== Cricketers, including biographical dictionaries and collections of biographies *[[w:ESPNcricinfo|ESPNcricinfo]] *[[w:CricketArchive|CricketArchive]] *John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ] *World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ] *Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ] *Sproat. Debrett's Cricketers' Who's Who. 1980. *S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934. *Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ] *Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ] *Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC] *Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ] Gloucestershire *Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html] *Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C]. *Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795] Australia *The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ] *Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ] *C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894. *Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus & Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ]. ==Geology== *Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History. ==Mineralogy== *Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&pg=PA473#v=onepage&q&f=false 473] to 476. *Battey, Maurice Hugh. Mineralogy for students. Oliver & Boyd. 1972. 2nd Ed. Longman. 1981. ==Paper== See [[s:Category:Paper]] *Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf] *West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog] ==Books== *British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ] *Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ] *Book News. 1882 to 1918. (John Wanamaker). Called "Book News Monthly" from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&pg=PP7#v=onepage&q&f=false] *Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ] *U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ] *Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ] *Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ] *Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C] *Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ] Rare *Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ] Printed *Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&pg=PP1#v=onepage&q&f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&pg=PP1#v=onepage&q&f=false vol 31] ==Paperback and Paperbound== *Swados, "Paper Books: What do they Promise?" (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114 *Wagman, "The Paperbound Book Business" (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&pg=RA4-PA9#v=onepage&q&f=false 9] (No 5, November) ==Languages== Maltese *See [[w:mt:Bibljografija tal-lingwa Maltija]] Judaeo-Spanish (Ladino) *See [[w:lad:Vikipedya:Bibliografia del djudeo-espanyol]] ==Science== *Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998. ==Entertainment== *The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/] ==Television== *Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&pg=PA1#v=onepage&q&f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television] Magazines *The Radio Times *TV Times Newspaper television reviews etc United Kingdom *A A Gill. Paper View: The Best of the Sunday Times Television Columns. *"Choice" or "Television and Radio Choice" in "Television and Radio". 1991. Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column. *"Choice" or "TV Choice" in "Television and Radio". The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the "Life & Times" section of the newspaper, for issues of the newspaper where "Life & Times" is a separate section. Otherwise they are printed in the middle of newspaper. *"Choice" or "TV Choice" in "Television and Radio". The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column. *"Choice". The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper. *"Television Choice". The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper. *"Review". The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper. *There are reviews in: **The Independent, The Guardian, The Financial Times, and The Daily Telegraph Netherlands *"TV: Films Video" in "televisie en radio woensdag". Limburgs Dagblad. *"show". Limburgs Dagblad. Japan *"Today's Choice" in "TV/Radio". The Japan Times. Music *Tele-Tunes Archives and listings *[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables]. ==Cinema== *Edgar Anstey, "The Cinema" (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes "Review of the Year". ==Animation== *John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005. *Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&pg=PA1958#v=onepage&q&f=false]. ==Colours== *Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/] ==Culture== *Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&pg=PP1#v=onepage&q&f=false] *Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&pg=PP1#v=onepage&q&f=false] *Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&pg=PP1#v=onepage&q&f=false] *Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular culture *Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue). ==Bilateral== Britain and Japan *Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ] ==Prehistoric life== Prehistoric animals *[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.] *Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&pg=PA269#v=onepage&q&f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&pg=PA5#v=onepage&q&f=false]. **Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977. Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ] **Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme. Dinosaurs *Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.] ==Continents== ===Asia=== ====Far East==== Bibliography *Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&pg=PP5#v=onepage&q&f=false] ==See also== *[[Bibliography]] ==Notes== {{Reflist}} {{subpagesif}} [[Category:Bibliographies]] [[Category:Research]] mkw1g0zk9rk19f2dsfn2f6tfrl345qx Motivation and emotion/Book/2015/Schadenfreude 0 202457 2803484 2324532 2026-04-08T06:43:51Z Jtneill 10242 /* What is schadenfreude? */ + image 2803484 wikitext text/x-wiki {{title|Schadenfreude:<br>Why do we feel pleasure in the suffering of others?}} {{MECR3|1=http://prezi.com/mzqkmuvaf-ii/?utm_campaign=share&utm_medium=copy}} __TOC__ ==Overview== Schadenfreude is a complex emotion that we feel when others suffer a misfortune. However, instead of feelings of sympathy, schadenfreude evokes feelings of joy and pleasure. Schadenfreude has been considered immoral and malicious, and is closely linked to envy, one of the seven biblical sins (Takahashi, et al., 2009). For these reasons, many have argued that schadenfreude is harmful to social relations (Heider, 1958). Other research has attempted to combat that idea, as they argue that schadenfreude is a healthy emotion, despite the fact that it is not always appropriate or polite to share it with others (Spurgin, 2015). Understanding what schadenfreude is, where it comes form in terms of psychological theories, and why we encounter feelings of schadenfreude, will help to understand and improve on our emotional lives. == Schadenfreude == === What is schadenfreude? === [[File:Schadenfreude.png|''Figure 1''. An artificially generated image of the facial expression of schadenfreude]] Schadenfreude is a German word which translates to the pleasure which is derived from the misfortune of others (Leach, Spears, Branscombe, & Doosje, 2003). Heider (1958) discussed how schadenfreude is a malicious emotion as it is an incongruous reaction to anothers'{{grammar}} misfortune. Heider (1958) is saying that instead of being sympathetic when another person is suffering, which could be considered the socially acceptable response, feelings of pleasure are seen as taboo and immoral (Leach, 2003). This feeling is typically seen as shameful or as a moral failing (Spurgin, 2015). Many people hide their feelings of schadenfreude, and many may not even realise that they are feeling pleasure at others{{grammar}} misfortune. This can stem from things such as gloating or joy at your basketball team winning a game. Both have emotions of schadenfreude behind them. Schadenfreude has its roots in [[w:Social_comparison_theory|Social Comparison Theory]]. This theory, largely influenced by Festinger (1954), states that we evaluate our abilities and opinions by comparing our views with others, and that we want people in similar groups to like us, so will change our wants and beliefs to match theirs (Myers, 2014). Myers (2014) also describes social comparison as evaluating our abilities and opinions by comparing ourselves to others. As schadenfreude is a social comparison, where you are comparing yourself against the misfortune of someone else, you are forming an opinion or judging your own abilities on the others{{grammar}} misfortune. Schadenfreude is a complex cognitive emotion that has many different reasons as to why we feel it (Reeve, 2015). Schadenfreude can be derived from feelings of envy, instability in ones'{{grammar}} self-worth, personal gain, when it is believed that the misfortune is deserved, along with biological factors (see Figure 1). == External Activities == === Video === To see an example of Pleasure derived from others misfortune follow this link to YouTube [https://www.youtube.com/watch?v=mcRyTdFKjPU 14 awesome viral video fails in 30 seconds] === Poll === After watching this video please complete this short poll to see how others feel when it comes to Schadenfreude [http://www.easypolls.net/poll.html?p=562c7466e4b09c75340b5249 Link to poll] Why do some people find videos like this funny? They could be experiencing feelings of schadenfreude, as they are getting pleasure from the suffering of others. But why do we feel this way? == Why do we feel pleasure in the suffering of others? == === The role of self-evaluation === When a person’s positive self-evaluation is threatened or harmed, they may have a strong motivation to protect or restore their self-evaluation (Van Dijk, 2013). One possible course of action to achieving this positive self-view can involve comparing one’s own situation to that of another person (Van Dijk, 2013). As a result, comparing another person’s misfortune may provide a sense of self worth or value to ones{{grammar}} own life. This means that people can use social comparisons and enjoy the misfortune of others as it provides a more positive self-evaluation. Research conducted by van Dijk, Ouwerkerk, Wesseling, & van Koninbruggen (2011) supports the idea that schadenfreude can be intensified by a threat to our self evaluation. They hypothesise that another reason for people to feel schadenfreude is because it satisfies their need to view themselves positively (van Dijk, 2011). This is argued in [[w:Social_comparison_theory|Social Comparison Theory]] which suggests that events and experiences that satisfy our concerns elicit positive emotions, whereas threat or harm will produce negative emotions (van Dijk, 2011). A way that people can make themselves feel better, according to [[w:Social_comparison_theory|Social Comparison Theory]], is to compare themselves to those who are less fortunate, also called 'downward social comparison' (van Dijk, 2011). Therefore, it is possible to argue that those who are suffering from self evaluation threat (and experiencing negative emotions), will use downward social comparison to help elicit positive emotions (Wills, 1981). The aim of Van Dijk's and his colleagues' (2011) research was to demonstrate that self-evaluation threat intensified schadenfreude in both threat-related and threat-unrelated domains. They were able to find that a threat to self-evaluation caused higher feelings of schadenfreude, and this was also possible to provoke in a threat-unrelated domain. This shows that self-evaluation can play a role towards feelings of schadenfreude. [[File:Children marbles.jpg|thumb|Image 2. ''Envy shown in children with marbles'']] === Envy === Envy had contradicting results when it came to schadenfreude. Many argued that there was a link between schadenfreude and envy, while others argued against this. van Dijk et al. (2006) investigated these contradictory results, and found that there is a link between schadenfreude and envy, but only when the misfortune fell upon someone who had some basis of similarity (e.g., gender). There{{grammar}} results found that if participants learnt about a misfortune of the opposite gender, schadenfreude would not be experienced (van Dijk, et al., 2006). However, when the same gender was identified as suffering misfortune, schadenfreude was identified. Smith, Powell, Combs, and Schurtz (2009) also show the correlation between envy and schadenfreude. They claim that envy is the polar opposite of a downward social comparison (Heider, 1958), however, when a misfortune occurs to someone who is envied, it transformed the comparison to a downwards one (Smith, et al., 2009). Conflicting reports on whether schadenfreude and envy are linked have been found, yet Smith et al., (2009), were able to replicate results where students who were enviable of another student felt greater schadenfreude when the person they envied suffered a misfortune, compared with those who were not in the envy group in the experiment. This provides empirical evidence that envy can lead to increased feelings of schadenfreude (Smith, et al., 2009). Smith et al., (2009) continue to remark that superiority to others does not always lead to envy, but when it does, this greatly increases the likelihood of schadenfreude. === In-Group Inferiority === An in-group refers to when an individual will recognise themselves as part of a group when they identify with them on some sort of level. For example, when someone identifies with a sporting group e.g. a football team, they begin to become part of the in-group. Another example is when people associate themselves with their university, an in-group forms. In-group inferiority refers to how people can feel pleasure at the misfortune of others in an in-group situation. For example, when a football team wins, that group will feel a sense of joy at the misfortune of the other team. Smith et al. (2009) suggests that when people identify with a group, the group becomes part of the individual, and the individual becomes part of the group. Leach et al. (2003) argue that schadenfreude is only evident when a third party or situation is the one that causes the misfortune, meaning that schadenfreude cannot occur if the pleasure is experienced when you are the cause of another persons misfortune. They suggest that schadenfreude should increase when an outer-group suffer misfortune in an area of high interest to the in-group members. They also delve into the idea that in-group inferiority will increase feelings of schadenfreude (Leach, et al., 2003; Leach and Spears, 2009). Leach et al. (2003) were successful in showing that when an individual felt more passionately about what formed the group (e.g. football) higher levels of schadenfreude were evoked when a third party suffered misfortune (e.g. lost a football match), and those who were less passionate, yet still were associated with the in-group had lesser feelings of schadenfreude. They were also able to demonstrate that schadenfreude was increased when feelings of in-group inferiority were experienced, however, this only affected those with lower interests (Leach, et al., 2003). Leach et al. (2003) also express that the threat to in-group inferiority and the increase in schadenfreude to those with higher interests was not seen, as those who had higher interests were already experiencing higher levels of schadenfreude. === Personal Gain (Competition) === Smith, et al. (2009) argue that the emotion of schadenfreude can be a result of a personal gain. They liken this to competition, where when you, or your team wins, you feel pleasure and this is ultimately in the suffering of the other team (Smith, et al., 2009). This idea of competition is seen in other aspects of life, and more often in day to day situations. It is arguably under-appreciated as to how often schadenfreude appears in a competitive everyday situation (Smith, et al., 2009). For example, if you are up for a new job, there is most likely going to be more than one person up for the position, and if you are successful in the process, you will most likely feel joy. This feeling of schadenfreude is one that is less ugly compared to other feelings derived from other places such as envy. A competitive nature is somewhat highly regarded (as seen with our tendency to highlight sports, and sports people), and seen to be quite natural (Smith, et al., 2009). Smith et al., (2009) discuss how this idea of personal gain is also evident in politics. Combs, Powell, Schurtz, and Smith (2009), conducted an experiment in the United States where they assessed whether schadenfreude was felt with political associations. They tested this by primarily assessing students{{grammar}} political identification, then by asking them to read an article which made out something embarrassing (or unfortunate) about the party leaders for both their party and the opposing party (Combs, et al., 2009). They found that schadenfreude was present when participants were shown articles about the opposing parties, and that the level of schadenfreude found depended on how affiliated one was with their{{grammar}} political party (Combs, et al., 2009). This also ties in with the idea of in-group identification, as these examples of schadenfreude are mostly group based successes or failures. They still hold the idea that when your group wins, you feel pleasure - at the misfortune of others. These findings emphasize the fact that schadenfreude is much more common than we would like to admit, is found in everyday life, and it is often regarded as natural and praised (Smith, et al., 2009). === Deserved Misfortune === Another justification for schadenfreude is the sense of deserved misfortune. When we feel that the misfortune that one has suffered is deserved, a feeling of pleasure is derived. It is argued that the feeling of deserved misfortune, which creates the feeling of schadenfreude, is a form of karmic retribution and gives us a sense of equilibrium (Lerner, & Miller, 1978). van Dijk, Ouwerkerk, Goslinga, & Nieweg (2005) showed the first empirical evidence on deserved misfortune and its link to schadenfreude. They showed evidence of schadenfreude increasing when it was perceived that the misfortune was more deserved (van Dijk, et al., 2005). They used a manipulation of responsibility to obtain differences in deserved misfortune, which led to the evidence that schadenfreude and a feeling of deserved misfortune are linked (van Dijk et al., 2005). Smith et al., (2009) discuss how this is also linked with hypocrisy. They explain that when we feel someone has been a hypocrite, we feel pleasure in the form of schadenfreude, at their misfortune. This is because we believe that the misfortune they are suffering is deserved. Smith, et al., (2009) yielded results in an experiment to examine the links between schadenfreude, deserved misfortune, and hypocrisy. They asked participants to read an article that presented an interview with a fellow student, where the student was either part of a campus organization about increasing academic integrity (high hypocrisy) or a student who was part of a French club (low hypocrisy) (Smith,et al., 2009). Participants were then shown a second article which said that the fellow student (in either case) was caught for plagiarism (Smith, et al., 2009). The results showed that those who were in the high hypocrisy group showed higher feelings of schadenfreude and that the student deserved the misfortune in comparison to those who were in the low hypocrisy group (Smith, et al. 2009). Smith, et al. (2009) also found similar outcomes when they changed the first article to be the same for all participants, and the manipulation came in the second article, where the other student was either caught in an immoral action that either matched the initial action that they were fighting against, or something completely unrelated. The results showed that when the immoral action matched that of the initial action higher levels of deserved misfortune and schadenfreude were felt. Unfortunately, this exact study was never published on its own, which questions whether there were problems with integrity in the research. Pietraszkiewicz (2013) discusses how schadenfreude and deserved misfortune are correlated to a just world belief. It was found that a threat to ones{{grammar}} just world belief increased ones pleasure at anothers' misfortune. Pietraszkiewicz (2013) argues that when failure is deserved, the greater the responsibility of the failure is, therefore, more schadenfreude is felt. === Biological components === ==== The role of Oxytocin ==== Research that was conducted by Shamay-Tsoory, et al. (2009) investigated the role of oxytocin in envy and gloating, which are both related to schadenfreude. Shamay-Tsoory, et al. (2009) discuss how oxytocin (a peptide hormone) has been shown to have implications in the social behaviour of humans and mammals. Many of the research into oxytocin looks at maternal behaviours such as contraction regulation in labour, as well as parental behaviours like trusting collaborators. They suggest that previous research has shown that oxytocin release is related to pro-social behaviours (Sharmay-Tsoory, et al., 2009). Seeing as pro-social behaviours are increased by oxytocin, your negative social behaviours, like envy and gloating, would logically be reduced. However, there has been an indication that this is not the case (Sharmay-Tsoory, et al., 2009). Sharmay-Tsoory et al. (2009) conducted an experiment which looked at the increase of oxytocin levels and its effect on these negatively perceived behaviours, such as schadenfreude. They concluded that gloating and envy, or schadenfreude, showed significantly higher rates of these emotions when oxytocin was given (Sharmay-Tsoory, et al., 2009). This research has provided evidence that oxytocin increases varying behaviours which are related to social behaviour, which in many roles are associated with parenting. ==== Neural Correlates ==== [[File:MRI anterior cingulate.png|alt=|thumb|Image 3. ''MRI of anterior cingulate cortex'' ]] [[File:Dopamine Pathways.png|alt=|thumb|Image 4. ''Position of the Ventral Striatum'']] Envy and schadenfreude are related emotions. Takahashi, et al. (2009) looked at the areas of the brain that were active when feelings of envy and schadenfreude were evoked. Using functional magnetic resonance imaging (fMRI) researchers looked for activity in the dorsal anterior cingulate cortex (dACC) (seen in image 2.) when envy was felt, as the anterior cingulate cortex is the area that is activated when our positive self-concept is being conflicted with external information, social pain, or cognitive conflicts (Takahashi, et al., 2009). When investigating the emotion of schadenfreude they were looking for activation in the ventral striatum, which is the central node of the rewards processing area (Takahashi, et al., 2009). The reward would be the joy that is derived in schadenfreude. Takahashi, et al. (2009) found that both areas which were targeted in their respective trials activated when the respective emotion was emitted. They found that when people had higher levels of schadenfreude, greater activation was seen in the ventral striatum. This was also found to be the case when investigating envy. Greater levels of envy showed higher activation in the dACC (Takahashi, et al., 2009). == Quiz == <quiz display="simple"> {What is Schadenfeude? |type="()"} - Freud's son + Pleasure derived from others misfortune - Pleasure derived from others fortune - Pain derived from others misfortune {What Peptide Hormione plays a role in Schadenfreude? |type="()"} - Oxycontin - glucocorticoids - Prolactin + Oxytocin {Which of the following does NOT play a role in Schadenfreude? |type="()"} - Envy - Self-worth + Anhedonia - Deserved misfortune {Which area of the brain is stimulated when you feel the emotion of schadenfreude? |type="()"} - dorsal Anterior Cingulate Cortex + Ventral Striatum - Prefrontal Cortex - Subgenual Cingulate </quiz> ==Conclusion == Schadenfreude is a complex emotion which can have many levels (van Dijk, 2011). It has many underlying concepts which relate to Social Comparison Theory, especially when evaluating the role of self-evaluation and schadenfreude. It is more commonly seen as a morally disturbed emotion, especially when feelings of envy, self-evaluation, and in-group inferiority are causes. However, it is seen quite commonly in everyday life, especially when it comes to situations or individuals with a competitive nature. There are many layers that underlie why schadenfreude occurs. Emotions such as envy, feelings of deservingness, personal gain, in-group inferiority and self-evaluation can all play a role in schadenfreude and why we feel this emotion, and there can be more than one reason as to why we experience schadenfreude. Other biological reasons, such as the role of oxytocin, and activity in the ventral striatum also play a role in feelings of schadenfreude. Schadenfreude is difficult to elicit in a clinical setting in an ethical way, which may be why conflicting results have been obtained for much of the research. Limitations in assessing schadenfreude may include social biases, as schadenfreude is an emotion that is generally considered immoral. This can lead to participants under-reporting their feelings of schadenfreude when being asked about it. A possible solution to this would be to conduct the schadenfreude eliciting part of the experiment under fMRI, as activity in the ventral striatum has been linked to schadenfreude, and it may be another way to see if someone is experiencing schadenfreude, without self-reporting. This may become more costly and less time effective, which would be a reason to stay clear of this research technique, however, it does become an option. Due to the many aspects and layers of schadenfreude, van Dijk et al., (2011) suggest that future research into which determinants effect schadenfreude under what circumstances will aid in further understanding why we feel pleasure in others misfortune. There may possibly be other reasons as to why we feel schadenfreude, greater research into the biological aspects of schadenfreude may also enhance our understanding of this emotion. Through investigating schadenfreude and the reasons why we might feel this emotion can aid to enrich our understanding and improve on out emotional lives, as when an experience of schadenfreude is likely to occur or is experienced, taking a step back and evaluating why we are feeling this emotion may lead to a greater emotional understanding and awareness. ==See also== [[Motivation and emotion/Book/2013/Deservingness and emotion#Schadenfreude|Deservingness and Emotion]] - Motivation and Emotion 2013 [[Motivation and emotion/Book/2011/Envy|Envy]] - Motivation and Emotion 2011 [[w:Schadenfreude|Schadenfreude]] - Wikipedia ==References== {{Hanging indent|1= Combs, D. J. Y., Powell, C. A. J., Schurtz, D. R., & Smith, R. H. (2009). Politics, Schadenfreude, and in-group identification: the sometimes happy thing about poor economy and death. ''Journal of Experimental Psychology, 45(4)'', 635-646. doi: 10.1016/j.jesp.2009.02.009 Festinger, L. (1954). A theory of social comparison processes. Human relations, 7(2), 117-140. Heider, F. (1958). ''The Psychology of Interpersonal Relations''. New York: Wiley Leach, C. W., & Spears, R. (2009). Dejection at in-group defeat and schadenfreude toward second- and third- party out-groups. ''Emotion, 9(5)'', 659-665. doi: 10.1037/a0016815 Leach, C. W., Spears, R., Branscombe, N. R., & Doosje, B. (2003). Malicious pleasure: schadenfreude at the suffering of another group. ''Journal of Personality and Social Psychology'', ''84(5),'' 932-943. doi: 10.1037/0022-3514.84.5.932 Lerner, M. J., & Miller, D. T. (1978). Just world research and the attribution processes: looking back and ahead. ''Psychological Bulletin, 85(8),'' 1030-1051. doi: 10.1037/0033-2909.85.5.1030 Louis, W. (2014). Group Influence. In Myer, D. G. (Eds.), ''Social Psychology'' (287). North Ride, N.S.W.: McGraw-Hill Pietraszkiewicz. A. (2013). Schdenfeude and just world belief. ''Australian Journal of Psychology, 65'', 188-194. doi: 10.1111/ajpy.12020 Reeve, J. (2015). ''Understanding motivation and emotion'' (6th ed.). Hoboken, NJ: Wiley. Shamay-Tsoory, S. G,. Fischer, M., Dvash, J., Harari, H., Perach-Bloom, N., & Levkovitz, Y. (2009). Intranasal administration of oxytocin increases envy and schadenfreude (gloating). ''Biological Psychiatry, 66(8)'', 864-870. doi: 10.1016/j.biopsych.2009.06.009 Smith, R. H., Powell, C. A. J., Combs, D. J. Y., & Schurtz. D. R. (2009). Exploring the when and why of schadenfreude. ''Social and Personality Psychology Compass, 3(4)'', 530-546. doi: 10.1111/j.1751-9004.2009.00181.x Spurgin, E. (2015). An emotional-freedom defense of schadenfreude. ''Ethical Theory and Modern Practice, 18'', 767-784. doi: 10.1007/s10677-014-9550-8 Van Dijk, W. W. (2013). Why do we sometimes enjoy the misfortune of others? ''The Inquisitive Mind'' Retrieved from: http://www.in-mind.org/blog/post/why-do-we-sometimes-enjoy-the-misfortune-of-others van Dijk, W. W., Goslinga, O. S., Nieweg, M., & Gallucci, M. (2006). When people fall from grace: reconsidering the role of envy in schadenfreude. ''Emotion, 6(1)'', 156-160. doi: 10.1037/1528-.3542.6.1.156 van Dijk, W. W., Ouwerkerk, J., Goslinga, S., & Nieweg, M. (2005). Deservingness and Schadenfreude. ''Cognition and Emotion, 19(6),'' 933-939. doi: 10.1080/02699930541000066 van Dijk, W. W., Ouwerkerk, J., Wesseling, Y. M., & van Koningsbruggen, G. M. (2011). Towards understanding pleasure at the misfortunes of others: the impact of self-evaluation threat on schadenfreude. ''Cognition and Emotion'', 25(2), 360-368. doi: 10.1080/02699931.2010.487365 Wills, T. A. (1981). Downward comparison principles in social psychology. ''Psychological Bulletin, 90(2),'' 245-271. }} ==External links== [http://www.livescience.com/17398-schadenfreude-affirmation.html Schadenfreude Explained: Why We Secretly Smile When Others Fail] [http://www.wsj.com/articles/schadenfreude-is-in-the-zeitgeist-but-is-there-an-opposite-term-1434129186 Schadenfreude Is in the Zeitgeist, but Is There an Opposite Term?] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Schadenfreude]] oj9y6yic8a3e4alb83xrqyu7yelcsg0 2803485 2803484 2026-04-08T06:44:14Z Jtneill 10242 /* What is schadenfreude? */ 2803485 wikitext text/x-wiki {{title|Schadenfreude:<br>Why do we feel pleasure in the suffering of others?}} {{MECR3|1=http://prezi.com/mzqkmuvaf-ii/?utm_campaign=share&utm_medium=copy}} __TOC__ ==Overview== Schadenfreude is a complex emotion that we feel when others suffer a misfortune. However, instead of feelings of sympathy, schadenfreude evokes feelings of joy and pleasure. Schadenfreude has been considered immoral and malicious, and is closely linked to envy, one of the seven biblical sins (Takahashi, et al., 2009). For these reasons, many have argued that schadenfreude is harmful to social relations (Heider, 1958). Other research has attempted to combat that idea, as they argue that schadenfreude is a healthy emotion, despite the fact that it is not always appropriate or polite to share it with others (Spurgin, 2015). Understanding what schadenfreude is, where it comes form in terms of psychological theories, and why we encounter feelings of schadenfreude, will help to understand and improve on our emotional lives. == Schadenfreude == === What is schadenfreude? === [[File:Schadenfreude.png|thumb|200px|''Figure 1''. An artificially generated image of the facial expression of schadenfreude]] Schadenfreude is a German word which translates to the pleasure which is derived from the misfortune of others (Leach, Spears, Branscombe, & Doosje, 2003). Heider (1958) discussed how schadenfreude is a malicious emotion as it is an incongruous reaction to anothers'{{grammar}} misfortune. Heider (1958) is saying that instead of being sympathetic when another person is suffering, which could be considered the socially acceptable response, feelings of pleasure are seen as taboo and immoral (Leach, 2003). This feeling is typically seen as shameful or as a moral failing (Spurgin, 2015). Many people hide their feelings of schadenfreude, and many may not even realise that they are feeling pleasure at others{{grammar}} misfortune. This can stem from things such as gloating or joy at your basketball team winning a game. Both have emotions of schadenfreude behind them. Schadenfreude has its roots in [[w:Social_comparison_theory|Social Comparison Theory]]. This theory, largely influenced by Festinger (1954), states that we evaluate our abilities and opinions by comparing our views with others, and that we want people in similar groups to like us, so will change our wants and beliefs to match theirs (Myers, 2014). Myers (2014) also describes social comparison as evaluating our abilities and opinions by comparing ourselves to others. As schadenfreude is a social comparison, where you are comparing yourself against the misfortune of someone else, you are forming an opinion or judging your own abilities on the others{{grammar}} misfortune. Schadenfreude is a complex cognitive emotion that has many different reasons as to why we feel it (Reeve, 2015). Schadenfreude can be derived from feelings of envy, instability in ones'{{grammar}} self-worth, personal gain, when it is believed that the misfortune is deserved, along with biological factors (see Figure 1). == External Activities == === Video === To see an example of Pleasure derived from others misfortune follow this link to YouTube [https://www.youtube.com/watch?v=mcRyTdFKjPU 14 awesome viral video fails in 30 seconds] === Poll === After watching this video please complete this short poll to see how others feel when it comes to Schadenfreude [http://www.easypolls.net/poll.html?p=562c7466e4b09c75340b5249 Link to poll] Why do some people find videos like this funny? They could be experiencing feelings of schadenfreude, as they are getting pleasure from the suffering of others. But why do we feel this way? == Why do we feel pleasure in the suffering of others? == === The role of self-evaluation === When a person’s positive self-evaluation is threatened or harmed, they may have a strong motivation to protect or restore their self-evaluation (Van Dijk, 2013). One possible course of action to achieving this positive self-view can involve comparing one’s own situation to that of another person (Van Dijk, 2013). As a result, comparing another person’s misfortune may provide a sense of self worth or value to ones{{grammar}} own life. This means that people can use social comparisons and enjoy the misfortune of others as it provides a more positive self-evaluation. Research conducted by van Dijk, Ouwerkerk, Wesseling, & van Koninbruggen (2011) supports the idea that schadenfreude can be intensified by a threat to our self evaluation. They hypothesise that another reason for people to feel schadenfreude is because it satisfies their need to view themselves positively (van Dijk, 2011). This is argued in [[w:Social_comparison_theory|Social Comparison Theory]] which suggests that events and experiences that satisfy our concerns elicit positive emotions, whereas threat or harm will produce negative emotions (van Dijk, 2011). A way that people can make themselves feel better, according to [[w:Social_comparison_theory|Social Comparison Theory]], is to compare themselves to those who are less fortunate, also called 'downward social comparison' (van Dijk, 2011). Therefore, it is possible to argue that those who are suffering from self evaluation threat (and experiencing negative emotions), will use downward social comparison to help elicit positive emotions (Wills, 1981). The aim of Van Dijk's and his colleagues' (2011) research was to demonstrate that self-evaluation threat intensified schadenfreude in both threat-related and threat-unrelated domains. They were able to find that a threat to self-evaluation caused higher feelings of schadenfreude, and this was also possible to provoke in a threat-unrelated domain. This shows that self-evaluation can play a role towards feelings of schadenfreude. [[File:Children marbles.jpg|thumb|Image 2. ''Envy shown in children with marbles'']] === Envy === Envy had contradicting results when it came to schadenfreude. Many argued that there was a link between schadenfreude and envy, while others argued against this. van Dijk et al. (2006) investigated these contradictory results, and found that there is a link between schadenfreude and envy, but only when the misfortune fell upon someone who had some basis of similarity (e.g., gender). There{{grammar}} results found that if participants learnt about a misfortune of the opposite gender, schadenfreude would not be experienced (van Dijk, et al., 2006). However, when the same gender was identified as suffering misfortune, schadenfreude was identified. Smith, Powell, Combs, and Schurtz (2009) also show the correlation between envy and schadenfreude. They claim that envy is the polar opposite of a downward social comparison (Heider, 1958), however, when a misfortune occurs to someone who is envied, it transformed the comparison to a downwards one (Smith, et al., 2009). Conflicting reports on whether schadenfreude and envy are linked have been found, yet Smith et al., (2009), were able to replicate results where students who were enviable of another student felt greater schadenfreude when the person they envied suffered a misfortune, compared with those who were not in the envy group in the experiment. This provides empirical evidence that envy can lead to increased feelings of schadenfreude (Smith, et al., 2009). Smith et al., (2009) continue to remark that superiority to others does not always lead to envy, but when it does, this greatly increases the likelihood of schadenfreude. === In-Group Inferiority === An in-group refers to when an individual will recognise themselves as part of a group when they identify with them on some sort of level. For example, when someone identifies with a sporting group e.g. a football team, they begin to become part of the in-group. Another example is when people associate themselves with their university, an in-group forms. In-group inferiority refers to how people can feel pleasure at the misfortune of others in an in-group situation. For example, when a football team wins, that group will feel a sense of joy at the misfortune of the other team. Smith et al. (2009) suggests that when people identify with a group, the group becomes part of the individual, and the individual becomes part of the group. Leach et al. (2003) argue that schadenfreude is only evident when a third party or situation is the one that causes the misfortune, meaning that schadenfreude cannot occur if the pleasure is experienced when you are the cause of another persons misfortune. They suggest that schadenfreude should increase when an outer-group suffer misfortune in an area of high interest to the in-group members. They also delve into the idea that in-group inferiority will increase feelings of schadenfreude (Leach, et al., 2003; Leach and Spears, 2009). Leach et al. (2003) were successful in showing that when an individual felt more passionately about what formed the group (e.g. football) higher levels of schadenfreude were evoked when a third party suffered misfortune (e.g. lost a football match), and those who were less passionate, yet still were associated with the in-group had lesser feelings of schadenfreude. They were also able to demonstrate that schadenfreude was increased when feelings of in-group inferiority were experienced, however, this only affected those with lower interests (Leach, et al., 2003). Leach et al. (2003) also express that the threat to in-group inferiority and the increase in schadenfreude to those with higher interests was not seen, as those who had higher interests were already experiencing higher levels of schadenfreude. === Personal Gain (Competition) === Smith, et al. (2009) argue that the emotion of schadenfreude can be a result of a personal gain. They liken this to competition, where when you, or your team wins, you feel pleasure and this is ultimately in the suffering of the other team (Smith, et al., 2009). This idea of competition is seen in other aspects of life, and more often in day to day situations. It is arguably under-appreciated as to how often schadenfreude appears in a competitive everyday situation (Smith, et al., 2009). For example, if you are up for a new job, there is most likely going to be more than one person up for the position, and if you are successful in the process, you will most likely feel joy. This feeling of schadenfreude is one that is less ugly compared to other feelings derived from other places such as envy. A competitive nature is somewhat highly regarded (as seen with our tendency to highlight sports, and sports people), and seen to be quite natural (Smith, et al., 2009). Smith et al., (2009) discuss how this idea of personal gain is also evident in politics. Combs, Powell, Schurtz, and Smith (2009), conducted an experiment in the United States where they assessed whether schadenfreude was felt with political associations. They tested this by primarily assessing students{{grammar}} political identification, then by asking them to read an article which made out something embarrassing (or unfortunate) about the party leaders for both their party and the opposing party (Combs, et al., 2009). They found that schadenfreude was present when participants were shown articles about the opposing parties, and that the level of schadenfreude found depended on how affiliated one was with their{{grammar}} political party (Combs, et al., 2009). This also ties in with the idea of in-group identification, as these examples of schadenfreude are mostly group based successes or failures. They still hold the idea that when your group wins, you feel pleasure - at the misfortune of others. These findings emphasize the fact that schadenfreude is much more common than we would like to admit, is found in everyday life, and it is often regarded as natural and praised (Smith, et al., 2009). === Deserved Misfortune === Another justification for schadenfreude is the sense of deserved misfortune. When we feel that the misfortune that one has suffered is deserved, a feeling of pleasure is derived. It is argued that the feeling of deserved misfortune, which creates the feeling of schadenfreude, is a form of karmic retribution and gives us a sense of equilibrium (Lerner, & Miller, 1978). van Dijk, Ouwerkerk, Goslinga, & Nieweg (2005) showed the first empirical evidence on deserved misfortune and its link to schadenfreude. They showed evidence of schadenfreude increasing when it was perceived that the misfortune was more deserved (van Dijk, et al., 2005). They used a manipulation of responsibility to obtain differences in deserved misfortune, which led to the evidence that schadenfreude and a feeling of deserved misfortune are linked (van Dijk et al., 2005). Smith et al., (2009) discuss how this is also linked with hypocrisy. They explain that when we feel someone has been a hypocrite, we feel pleasure in the form of schadenfreude, at their misfortune. This is because we believe that the misfortune they are suffering is deserved. Smith, et al., (2009) yielded results in an experiment to examine the links between schadenfreude, deserved misfortune, and hypocrisy. They asked participants to read an article that presented an interview with a fellow student, where the student was either part of a campus organization about increasing academic integrity (high hypocrisy) or a student who was part of a French club (low hypocrisy) (Smith,et al., 2009). Participants were then shown a second article which said that the fellow student (in either case) was caught for plagiarism (Smith, et al., 2009). The results showed that those who were in the high hypocrisy group showed higher feelings of schadenfreude and that the student deserved the misfortune in comparison to those who were in the low hypocrisy group (Smith, et al. 2009). Smith, et al. (2009) also found similar outcomes when they changed the first article to be the same for all participants, and the manipulation came in the second article, where the other student was either caught in an immoral action that either matched the initial action that they were fighting against, or something completely unrelated. The results showed that when the immoral action matched that of the initial action higher levels of deserved misfortune and schadenfreude were felt. Unfortunately, this exact study was never published on its own, which questions whether there were problems with integrity in the research. Pietraszkiewicz (2013) discusses how schadenfreude and deserved misfortune are correlated to a just world belief. It was found that a threat to ones{{grammar}} just world belief increased ones pleasure at anothers' misfortune. Pietraszkiewicz (2013) argues that when failure is deserved, the greater the responsibility of the failure is, therefore, more schadenfreude is felt. === Biological components === ==== The role of Oxytocin ==== Research that was conducted by Shamay-Tsoory, et al. (2009) investigated the role of oxytocin in envy and gloating, which are both related to schadenfreude. Shamay-Tsoory, et al. (2009) discuss how oxytocin (a peptide hormone) has been shown to have implications in the social behaviour of humans and mammals. Many of the research into oxytocin looks at maternal behaviours such as contraction regulation in labour, as well as parental behaviours like trusting collaborators. They suggest that previous research has shown that oxytocin release is related to pro-social behaviours (Sharmay-Tsoory, et al., 2009). Seeing as pro-social behaviours are increased by oxytocin, your negative social behaviours, like envy and gloating, would logically be reduced. However, there has been an indication that this is not the case (Sharmay-Tsoory, et al., 2009). Sharmay-Tsoory et al. (2009) conducted an experiment which looked at the increase of oxytocin levels and its effect on these negatively perceived behaviours, such as schadenfreude. They concluded that gloating and envy, or schadenfreude, showed significantly higher rates of these emotions when oxytocin was given (Sharmay-Tsoory, et al., 2009). This research has provided evidence that oxytocin increases varying behaviours which are related to social behaviour, which in many roles are associated with parenting. ==== Neural Correlates ==== [[File:MRI anterior cingulate.png|alt=|thumb|Image 3. ''MRI of anterior cingulate cortex'' ]] [[File:Dopamine Pathways.png|alt=|thumb|Image 4. ''Position of the Ventral Striatum'']] Envy and schadenfreude are related emotions. Takahashi, et al. (2009) looked at the areas of the brain that were active when feelings of envy and schadenfreude were evoked. Using functional magnetic resonance imaging (fMRI) researchers looked for activity in the dorsal anterior cingulate cortex (dACC) (seen in image 2.) when envy was felt, as the anterior cingulate cortex is the area that is activated when our positive self-concept is being conflicted with external information, social pain, or cognitive conflicts (Takahashi, et al., 2009). When investigating the emotion of schadenfreude they were looking for activation in the ventral striatum, which is the central node of the rewards processing area (Takahashi, et al., 2009). The reward would be the joy that is derived in schadenfreude. Takahashi, et al. (2009) found that both areas which were targeted in their respective trials activated when the respective emotion was emitted. They found that when people had higher levels of schadenfreude, greater activation was seen in the ventral striatum. This was also found to be the case when investigating envy. Greater levels of envy showed higher activation in the dACC (Takahashi, et al., 2009). == Quiz == <quiz display="simple"> {What is Schadenfeude? |type="()"} - Freud's son + Pleasure derived from others misfortune - Pleasure derived from others fortune - Pain derived from others misfortune {What Peptide Hormione plays a role in Schadenfreude? |type="()"} - Oxycontin - glucocorticoids - Prolactin + Oxytocin {Which of the following does NOT play a role in Schadenfreude? |type="()"} - Envy - Self-worth + Anhedonia - Deserved misfortune {Which area of the brain is stimulated when you feel the emotion of schadenfreude? |type="()"} - dorsal Anterior Cingulate Cortex + Ventral Striatum - Prefrontal Cortex - Subgenual Cingulate </quiz> ==Conclusion == Schadenfreude is a complex emotion which can have many levels (van Dijk, 2011). It has many underlying concepts which relate to Social Comparison Theory, especially when evaluating the role of self-evaluation and schadenfreude. It is more commonly seen as a morally disturbed emotion, especially when feelings of envy, self-evaluation, and in-group inferiority are causes. However, it is seen quite commonly in everyday life, especially when it comes to situations or individuals with a competitive nature. There are many layers that underlie why schadenfreude occurs. Emotions such as envy, feelings of deservingness, personal gain, in-group inferiority and self-evaluation can all play a role in schadenfreude and why we feel this emotion, and there can be more than one reason as to why we experience schadenfreude. Other biological reasons, such as the role of oxytocin, and activity in the ventral striatum also play a role in feelings of schadenfreude. Schadenfreude is difficult to elicit in a clinical setting in an ethical way, which may be why conflicting results have been obtained for much of the research. Limitations in assessing schadenfreude may include social biases, as schadenfreude is an emotion that is generally considered immoral. This can lead to participants under-reporting their feelings of schadenfreude when being asked about it. A possible solution to this would be to conduct the schadenfreude eliciting part of the experiment under fMRI, as activity in the ventral striatum has been linked to schadenfreude, and it may be another way to see if someone is experiencing schadenfreude, without self-reporting. This may become more costly and less time effective, which would be a reason to stay clear of this research technique, however, it does become an option. Due to the many aspects and layers of schadenfreude, van Dijk et al., (2011) suggest that future research into which determinants effect schadenfreude under what circumstances will aid in further understanding why we feel pleasure in others misfortune. There may possibly be other reasons as to why we feel schadenfreude, greater research into the biological aspects of schadenfreude may also enhance our understanding of this emotion. Through investigating schadenfreude and the reasons why we might feel this emotion can aid to enrich our understanding and improve on out emotional lives, as when an experience of schadenfreude is likely to occur or is experienced, taking a step back and evaluating why we are feeling this emotion may lead to a greater emotional understanding and awareness. ==See also== [[Motivation and emotion/Book/2013/Deservingness and emotion#Schadenfreude|Deservingness and Emotion]] - Motivation and Emotion 2013 [[Motivation and emotion/Book/2011/Envy|Envy]] - Motivation and Emotion 2011 [[w:Schadenfreude|Schadenfreude]] - Wikipedia ==References== {{Hanging indent|1= Combs, D. J. Y., Powell, C. A. J., Schurtz, D. R., & Smith, R. H. (2009). Politics, Schadenfreude, and in-group identification: the sometimes happy thing about poor economy and death. ''Journal of Experimental Psychology, 45(4)'', 635-646. doi: 10.1016/j.jesp.2009.02.009 Festinger, L. (1954). A theory of social comparison processes. Human relations, 7(2), 117-140. Heider, F. (1958). ''The Psychology of Interpersonal Relations''. New York: Wiley Leach, C. W., & Spears, R. (2009). Dejection at in-group defeat and schadenfreude toward second- and third- party out-groups. ''Emotion, 9(5)'', 659-665. doi: 10.1037/a0016815 Leach, C. W., Spears, R., Branscombe, N. R., & Doosje, B. (2003). Malicious pleasure: schadenfreude at the suffering of another group. ''Journal of Personality and Social Psychology'', ''84(5),'' 932-943. doi: 10.1037/0022-3514.84.5.932 Lerner, M. J., & Miller, D. T. (1978). Just world research and the attribution processes: looking back and ahead. ''Psychological Bulletin, 85(8),'' 1030-1051. doi: 10.1037/0033-2909.85.5.1030 Louis, W. (2014). Group Influence. In Myer, D. G. (Eds.), ''Social Psychology'' (287). North Ride, N.S.W.: McGraw-Hill Pietraszkiewicz. A. (2013). Schdenfeude and just world belief. ''Australian Journal of Psychology, 65'', 188-194. doi: 10.1111/ajpy.12020 Reeve, J. (2015). ''Understanding motivation and emotion'' (6th ed.). Hoboken, NJ: Wiley. Shamay-Tsoory, S. G,. Fischer, M., Dvash, J., Harari, H., Perach-Bloom, N., & Levkovitz, Y. (2009). Intranasal administration of oxytocin increases envy and schadenfreude (gloating). ''Biological Psychiatry, 66(8)'', 864-870. doi: 10.1016/j.biopsych.2009.06.009 Smith, R. H., Powell, C. A. J., Combs, D. J. Y., & Schurtz. D. R. (2009). Exploring the when and why of schadenfreude. ''Social and Personality Psychology Compass, 3(4)'', 530-546. doi: 10.1111/j.1751-9004.2009.00181.x Spurgin, E. (2015). An emotional-freedom defense of schadenfreude. ''Ethical Theory and Modern Practice, 18'', 767-784. doi: 10.1007/s10677-014-9550-8 Van Dijk, W. W. (2013). Why do we sometimes enjoy the misfortune of others? ''The Inquisitive Mind'' Retrieved from: http://www.in-mind.org/blog/post/why-do-we-sometimes-enjoy-the-misfortune-of-others van Dijk, W. W., Goslinga, O. S., Nieweg, M., & Gallucci, M. (2006). When people fall from grace: reconsidering the role of envy in schadenfreude. ''Emotion, 6(1)'', 156-160. doi: 10.1037/1528-.3542.6.1.156 van Dijk, W. W., Ouwerkerk, J., Goslinga, S., & Nieweg, M. (2005). Deservingness and Schadenfreude. ''Cognition and Emotion, 19(6),'' 933-939. doi: 10.1080/02699930541000066 van Dijk, W. W., Ouwerkerk, J., Wesseling, Y. M., & van Koningsbruggen, G. M. (2011). Towards understanding pleasure at the misfortunes of others: the impact of self-evaluation threat on schadenfreude. ''Cognition and Emotion'', 25(2), 360-368. doi: 10.1080/02699931.2010.487365 Wills, T. A. (1981). Downward comparison principles in social psychology. ''Psychological Bulletin, 90(2),'' 245-271. }} ==External links== [http://www.livescience.com/17398-schadenfreude-affirmation.html Schadenfreude Explained: Why We Secretly Smile When Others Fail] [http://www.wsj.com/articles/schadenfreude-is-in-the-zeitgeist-but-is-there-an-opposite-term-1434129186 Schadenfreude Is in the Zeitgeist, but Is There an Opposite Term?] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Schadenfreude]] msex1txljbef3qnuoz2yegvdgss5emi Digital Media Concepts 0 213879 2803443 2802777 2026-04-07T22:52:51Z Jaylor22 3061875 /* Spring 2026 */ 2803443 wikitext text/x-wiki == '''Digital Artists Wiki Project''' Assignment == === Assignment Description === For this assignment, you will write and publish a wiki-style article about a digital artist/technology of your choice on Wikiversity.<ref>https://www.techxlab.org/solutions/wikimedia-foundation-wikiversity</ref> You should begin by finding an article stub that needs to be developed or identifying a potential artist whose work is not featured currently. Remember that you are writing for the Wikiversity community and your article will be judged and graded on Wikipedia publishing standards.<ref>[[Wikipedia:Wikipedia:Policies and guidelines]]</ref> == Instructions == '''1. Create a user Profile here:'''<br> [[Special:CreateAccount]]<br> User Guidelines can be found here: [[Wikiversity:Introduction]]<br><br> '''2. Choose a Topic'''<br> You may choose any topic related to digital art and technology from any time period. Try to find a topic that's important to you and not already featured on wikipedia. In case you have difficulties finding a topic, please feel free to consult with your instructor.<br> Please note that each student must choose a unique topic. Once you have selected a digital artist, post your artist’s name along with your name under the Spring 2026 header. Please check that no one else has elected to work on the same artists. Example: <pre>* ~~~~[[Digital Media Concepts/YOUR PAGE TITLE]]</pre> for example <pre>* ~~~~[[Digital Media Concepts/Wafaa Bilal]]</pre> produces: * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] '''3. OR Create your page using the following:''' <inputbox> type=create width=80 preload={{FULLPAGENAME}}/Template editintro= buttonlabel=Create project subpage searchbuttonlabel= break=no prefix={{FULLPAGENAME}}/ placeholder=enter title here </inputbox><br> This template can help get you started: [[Digital_Media_Concepts/Template|Template]] == Links == A wikilink (or internal link) links a page to another page within English Wikipedia. In wikitext, links are enclosed in double square brackets like this: <pre>[[abc]]</pre> is seen as [[abc]] External links use absolute URLs to link directly to any web page. External links are enclosed in single square brackets (rather than double brackets as with internal links), with the optional link text separated from the URL by a space <pre>[http://www.example.org/ link text]</pre> will be rendered as: [http://www.example.org/ link text] When no link text is specified, external links appear numbered: <pre>[http://www.example.org/some-other-page]</pre> becomes: [http://www.example.org/some-other-page] Links with no square brackets display in their entirety <pre>http://www.example.org/</pre> displays as http://www.example.org === Spring 2026 === {{collapse top|Project List}} * [[User:Jaylor22|Jaylor22]] ([[User talk:Jaylor22|discuss]] • [[Special:Contributions/Jaylor22|contribs]]) 22:52, 7 April 2026 (UTC)[[Digital Media Concepts/Use of CGI in Motion Picture Film How to Train Your Dragon]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 02:21, 4 April 2026 (UTC)[[Digital Media Concepts/TikTok and Its Impact on Self-Esteem]] * [[User:Ksafa1|Ksafa1]] ([[User talk:Ksafa1|discuss]] • [[Special:Contributions/Ksafa1|contribs]]) 05:29, 1 April 2026 (UTC)[[Digital Media Concepts/The Rise of Digital Music Therapy Apps and Teen Mental Health]] * [[User:Mina Safa|Mina Safa]] ([[User talk:Mina Safa|discuss]] • [[Special:Contributions/Mina Safa|contribs]]) 03:59, 1 April 2026 (UTC)[[Digital Media Concepts/Digital Media Concepts/Algorithmic Mood Feeds]] * [[User:Aarna.Makam|Aarna.Makam]] ([[User talk:Aarna.Makam|discuss]] • [[Special:Contributions/Aarna.Makam|contribs]]) 00:08, 31 March 2026 (UTC)[[Digital Media Concepts/Virtual Concerts & Hologram-Style Live Events]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 18:27, 29 March 2026 (UTC)[[Digital Media Concepts/New StudyFetch]] * [[User:Vania Lat|Vania Lat]] ([[User talk:Vania Lat|discuss]] • [[Special:Contributions/Vania Lat|contribs]]) 20:33, 9 March 2026 (UTC) [[Digital Media Concepts/Technological Landscape of Esports]] * [[User:SarahLuber|SarahLuber]] ([[User talk:SarahLuber|discuss]] • [[Special:Contributions/SarahLuber|contribs]]) 20:59, 9 March 2026 (UTC)[[Digital Media Concepts/Notch's history In Games]] * [[User:5heep.doggydog|5heep.doggydog]] ([[User talk:5heep.doggydog|discuss]] • [[Special:Contributions/5heep.doggydog|contribs]]) 16:43, 13 March 2026 (UTC)[[Digital Media Concepts/Radio Telemetry in Bear Conservation]] * [[User:Qianqian Z|Qianqian Z]] ([[User talk:Qianqian Z|discuss]] • [[Special:Contributions/Qianqian Z|contribs]]) 05:19, 16 March 2026 (UTC) [[Digital Media Concepts/Zhang Yiming and Algorithm-Driven Content Platforms]] * [[User:EZMedina18|EZMedina18]] ([[User talk:EZMedina18|discuss]] • [[Special:Contributions/EZMedina18|contribs]]) 04:34, 27 March 2026 (UTC)[[Digital Media Concepts/Trackman Golf Launch Monitors]] * [[User:Triajj|Trianna J.]] ([[User talk:Triajj|discuss]] • [[Special:Contributions/Triajj|contribs]]) 23:26, 27 March 2026 (UTC)[[Digital Media Concepts/Digital Marketing Strategies in K-Pop]] * [[User:Qiurick|Qiurick]] ([[User talk:Qiurick|discuss]] • [[Special:Contributions/Qiurick|contribs]]) 20:50, 28 March 2026 (UTC)[[Digital Media Concepts/Simon Collins-Laflamme]] * [[User:SwarG07|SwarG07]] ([[User talk:SwarG07|discuss]] • [[Special:Contributions/SwarG07|contribs]]) 06:01, 30 March 2026 (UTC) [[Digital Media Concepts/Kanye West and the Living Album Concept]] * [[User:Joaquin A Quinonez|Joaquin A Quinonez]] ([[User talk:Joaquin A Quinonez|discuss]] • [[Special:Contributions/Joaquin A Quinonez|contribs]]) 20:25, 30 March 2026 (UTC)[[Digital Media Concepts/Environmental storytelling in Xenoblade Chronicles]] {{collapse bottom}} === Fall 2025 === {{collapse top|Project List}} * [[User:Yusuf152|Yusuf152] ([[User talk:Yusuf152|discuss]] • [[Special:Contributions/Yusuf152|contribs]]) 06:42, 27 October 2025 (UTC) [[Digital Media Concepts/Social Media Technology and the Arab Spring]] * [[User:Riaz23413|Riaz23413]] ([[User talk:Riaz23413|discuss]] • [[Special:Contributions/Riaz23413|contribs]]) 06:42, 27 October 2025 (UTC)[[Digital Media Concepts/Midjourney v6]]*[[User:JSDAID|JSDAID]] ([[User talk:JSDAID|discuss]] • [[Special:Contributions/JSDAID|contribs]]) 04:00, 26 October 2025 (UTC) [[Digital Media Concepts/Devine Lu Linvega]] * [[User:SrushDigitalMedia|SrushDigitalMedia]] ([[User talk:SrushDigitalMedia|discuss]] • [[Special:Contributions/SrushDigitalMedia|contribs]]) 00:23, 8 October 2025 (UTC)[[Digital Media Concepts/Jessica “Sting” Peterson]] *[[User:Lsingh29|Lsingh29]] ([[User talk:Lsingh29|discuss]] • [[Special:Contributions/Lsingh29|contribs]]) 06:47, 19 October 2025 (UTC)[[Digital Media Concepts/The Cultural Impact of Pokémon on Youth]] * [[User:Fnabong|Fnabong]] ([[User talk:Fnabong|discuss]] • [[Special:Contributions/Fnabong|contribs]]) 03:19, 24 October 2025 (UTC)[[Digital Media Concepts/Online Privacy Plan]] * [[User:Sharaf Obaid|Sharaf Obaid]] ([[User talk:Sharaf Obaid|discuss]] • [[Special:Contributions/Sharaf Obaid|contribs]]) 16:25, 25 October 2025 (UTC)[[Digital Media Concepts/Impact of AI on Social Media]] * [[User:A.A2711|A.A2711]] ([[User talk:A.A2711|discuss]] • [[Special:Contributions/A.A2711|contribs]]) 19:15, 25 October 2025 (UTC)[[Digital Media Concepts/Paul Allen's Philanthropic Practices]] * [[User:AdamFlick18|AdamFlick18]] ([[User talk:AdamFlick18|discuss]] • [[Special:Contributions/AdamFlick18|contribs]]) 06:19, 27 October 2025 (UTC)[[Digital Media Concepts/Automated Industrial Robotics Inc. (AIR)]] * [[User:Msingh132|Msingh132]] ([[User talk:Msingh132|discuss]] • [[Special:Contributions/Msingh132|contribs]]) 07:23, 27 October 2025 (UTC)[[Digital Media Concepts/The impact of renewable energy on emerging economies]] * [[User:CleaverCat2|<bdi>CleaverCat2</bdi>]] ([[User talk:CleaverCat2|discuss]] • [[Special:Contributions/CleaverCat2|contribs]]) 06:26, 26 October 2025 (UTC) '''[[Digital Media Concepts/Generative AI on Game Development]]''' * [[User:Wesleyphin|Wesleyphin]] ([[User talk:Wesleyphin|discuss]] • [[Special:Contributions/Wesleyphin|contribs]]) 23:04, 26 October 2025 (UTC)[[Digital Media Concepts/Sora 2]] {{collapse bottom}} === Spring 2025 === {{collapse top|Project List}} * [[User:Gsingh598|Gsingh598]] ([[User talk:Gsingh598|discuss]] • [[Special:Contributions/Gsingh598|contribs]]) 18:09, 10 April 2025 (UTC)[[Digital Media Concepts/Refik Anadol: Blending Data and Art Through Machine Learning]] * [[User:Safaiqbal11|Safaiqbal11]] ([[User talk:Safaiqbal11|discuss]] • [[Special:Contributions/Safaiqbal11|contribs]]) 04:58, 17 March 2025 (UTC)[[Digital Media Concepts/The Evolution of the iPhone and Its Cultural Impact]] * [[User:Weswu341|Weswu341]] ([[User talk:Weswu341|discuss]] • [[Special:Contributions/Weswu341|contribs]]) 01:27, 19 March 2025 (UTC)[[Digital Media Concepts/Yucef Merhi]] * [[User:Mdelacruz7|Mdelacruz7]] ([[User talk:Mdelacruz7|discuss]] • [[Special:Contributions/Mdelacruz7|contribs]]) 02:32, 25 March 2025 (UTC)[[Digital Media Concepts/Technology and its Influence on Romance.]] * [[User:Amojado1|Amojado1]] ([[User talk:Amojado1|discuss]] • [[Special:Contributions/Amojado1|contribs]]) 22:09, 28 March 2025 (UTC)[[Digital Media Concepts/Samir Vasavada]] * [[User:TianxinXu0201|TianxinXu0201]] ([[User talk:TianxinXu0201|discuss]] • [[Special:Contributions/TianxinXu0201|contribs]]) 19:15, 29 March 2025 (UTC)[[Digital Media Concepts/The Role of Autopilot in Tesla's Robotaxi Plan]] * [[User:Zsadozai|Zsadozai]] ([[User talk:Zsadozai|discuss]] • [[Special:Contributions/Zsadozai|contribs]]) 05:31, 30 March 2025 (UTC)[[Digital Media Concepts/The Influence of Live Creators]] * [[User:Dija8719|Dija8719]] ([[User talk:Dija8719|discuss]] • [[Special:Contributions/Dija8719|contribs]]) 06:53, 30 March 2025 (UTC)[[Digital Media Concepts/Smartphones Impact on Communication]] * [[User:Ksak2|Ksak2]] ([[User talk:Ksak2|discuss]] • [[Special:Contributions/Ksak2|contribs]]) 23:49, 30 March 2025 (UTC)[[Digital Media Concepts/The Reflection of the Nintendo Wii on the Nintendo Switch]] * [[User:Nyanoble|Nyanoble]] ([[User talk:Nyanoble|discuss]] • [[Special:Contributions/Nyanoble|contribs]]) 00:30, 31 March 2025 (UTC)[[Digital Media Concepts/ Impact AI has on the job market]] * [[User:Kjchai1|Kjchai1]] ([[User talk:Kjchai1|discuss]] • [[Special:Contributions/Kjchai1|contribs]]) 04:00, 31 March 2025 (UTC)[[Digital Media Concepts/Alex Karp's impact on the public view of AI]] * [[User:Mpiquero1]] 23:07, 30 March 2025 (UTC)[[Digital Media Concepts/Quantum Computing and Information Security]] * [[User:NoahMacias1|NoahMacias1]] ([[User talk:NoahMacias1|discuss]] • [[Special:Contributions/NoahMacias1|contribs]]) 04:15, 2 April 2025 (UTC)[[Digital Media Concepts/Marvel Rivals vs Overwatch]] * [[User:Cassidyhalen|Cassidyhalen]] ([[User talk:Cassidyhalen|discuss]] • [[Special:Contributions/Cassidyhalen|contribs]]) 10:28, 5 April 2025 (UTC)[[Digital Media Concepts/Waymo's impact on the economy]] * [[User:Glry2ua|Glry2ua]] ([[User talk:Glry2ua|discuss]] • [[Special:Contributions/Glry2ua|contribs]]) 06:50, 27 May 2025 (UTC)[[Digital Media Concepts/Counter-Drone Technologies]] {{collapse bottom}} === Fall 2024 === {{collapse top|Project List}} * [[User:Dawson Travis|Dawson Travis]] ([[User talk:Dawson Travis|discuss]] • [[Special:Contributions/Dawson Travis|contribs]]) 20:03, 2 November 2024 (UTC)[[Digital Media Concepts/Bruce Laval]] * [[User:Jillianrae247|Jillianrae247]] ([[User talk:Jillianrae247|discuss]] • [[Special:Contributions/Jillianrae247|contribs]]) 06:21, 28 October 2024 (UTC)[[Digital Media Concepts/David LaRochelle]] * [[User:Jmartin51|Jmartin51]] ([[User talk:Jmartin51|discuss]] • [[Special:Contributions/Jmartin51j|contribs]]) 22:46, 27 October 2024 (UTC)[[Creating Digital Media Concepts/Relationship of GPUs and Bitcoin Mining]] * [[User:Hero1ghj|Hero1ghj]] ([[User talk:Hero1ghj|discuss]] • [[Special:Contributions/Hero1ghj|contribs]]) 01:56, 28 October 2024 (UTC)[[Digital Media Concepts/The Fortnite Storyline]] * [[User:Catalinajordanvillar|Catalinajordanvillar]] ([[User talk:Catalinajordanvillar|discuss]] • [[Special:Contributions/Catalinajordanvillar|contribs]]) 06:57, 27 October 2024 (UTC)[[Digital Media Concepts/Evolution Of Disney Animation]] * [[User:Daid gem|Daid gem]] ([[User talk:Daid gem|discuss]] • [[Special:Contributions/Daid gem|contribs]]) 22:35, 25 October 2024 (UTC)[[Digital Media Concepts/Webtoon Adaptations]] * [[User:Hillary Cheng|Hillary Cheng]] ([[User talk:Hillary Cheng|discuss]] • [[Special:Contributions/Hillary Cheng|contribs]]) 18:37, 15 October 2024 (UTC)[[Digital Media Concepts/Lane Centering Technology]] * [[User:Chloecastellana|Chloecastellana]] ([[User talk:Chloecastellana|discuss]] • [[Special:Contributions/Chloecastellana|contribs]]) 23:04, 19 October 2024 (UTC)[[Digital Media Concepts/Jack Ma]] * [[User:Durontop7|Durontop7]] ([[User talk:Durontop7|discuss]] • [[Special:Contributions/Durontop7|contribs]]) 02:11, 28 October 2024 (UTC)[[Digital Media Concepts/Getting Over It with Bennett Foddy]] * [[User:Megamatt02001|Megamatt02001]] ([[User talk:Megamatt02001|discuss]] • [[Special:Contributions/Megamatt02001|contribs]]) 13:05, 28 October 2024 (UTC) [[Hideo Kojima's Podcast "Brain Structure"]] {{collapse bottom}} === Spring 2024 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 20:28, 22 April 2024 (UTC)[[Digital Media Concepts/Can't Help Myself Robot]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 23:24, 25 March 2024 (UTC)[[Digital Media Concepts/Online Shopping]] * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 03:44, 25 March 2024 (UTC)[[Digital Media Concepts/AI’s Impact on Art]] * [[User:Diegou04|Diegou04]] ([[User talk:Diegou04|discuss]] • [[Special:Contributions/Diegou04|contribs]]) 00:39, 25 March 2024 (UTC)[[Digital Media Concepts/Apple Vision Pro Environmental Considerations ]] * [[User:Omegajame456|Omegajame456]] ([[User talk:Omegajame456|discuss]] • [[Special:Contributions/Omegajame456|contribs]]) 10:40, 23 March 2024 (UTC)[[Digital Media Concepts/Smart Roads]] * [[User:Ashlyn45|Ashlyn45]] ([[User talk:Ashlyn45|discuss]] • [[Special:Contributions/Ashlyn45|contribs]]) 06:50, 22 March 2024 (UTC)[[Digital Media Concepts/Khan Academy Breakthrough]] * [[User:Myat K.S|Myat K.S]] ([[User talk:Myat K.S|discuss]] • [[Special:Contributions/Myat K.S|contribs]]) 10:10, 7 March 2024 (UTC)[[Digital Media Concepts/BILL GATES (William Henry Gates III)]] * [[User:Yuyani0915|Yuyani0915]] ([[User talk:Yuyani0915|discuss]] • [[Special:Contributions/Yuyani0915|contribs]]) 22:38, 13 March 2024 (UTC)[[Digital Media Concepts/Gaming Technology]] * [[User:RobbyDAID|RobbyDAID]] ([[User talk:RobbyDAID|discuss]] • [[Special:Contributions/RobbyDAID|contribs]]) 19:23, 19 March 2024 (UTC)[[Digital Media Concepts/Lost Cities]] * [[User:Isaiah Guerrero|Isaiah Guerrero]] ([[User talk:Isaiah Guerrero|discuss]] • [[Special:Contributions/Isaiah Guerrero|contribs]]) 04:15, 20 March 2024 (UTC)[[Digital Media Concepts/Nano Bots and CO2]] * [[User:Xavierrrrrrr01|Xavierrrrrrr01]] ([[User talk:Xavierrrrrrr01|discuss]] • [[Special:Contributions/Xavierrrrrrr01|contribs]]) 19:20, 20 March 2024 (UTC)[[Digital Media Concepts/Uses of the Raspberry Pi]] * [[User:Luna, D Media|Luna, D Media]] ([[User talk:Luna, D Media|discuss]] • [[Special:Contributions/Luna, D Media|contribs]]) 02:09, 23 March 2024 (UTC)[[Digital Media Concepts/Science S.T.E.M. Influencers]] * [[User:Yueminl|Yueminl]] ([[User talk:Yueminl|discuss]] • [[Special:Contributions/Yueminl|contribs]]) 22:39, 23 March 2024 (UTC)[[Digital Media Concepts/Information Cocoons Effect]] * [[User:Tarunya Dharma|Tarunya Dharma]] ([[User talk:Tarunya Dharma|discuss]] • [[Special:Contributions/Tarunya Dharma|contribs]]) 01:59, 24 March 2024 (UTC)[[Digital Media Concepts/AI Human Clones]] * [[User:GCanilao|GCanilao]] ([[User talk:GCanilao|discuss]] • [[Special:Contributions/GCanilao|contribs]]) 05:56, 24 March 2024 (UTC)[[Digital Media Concepts/Evolution of "Bits" in Game Graphics]] * [[User:Helldivers.2|Helldivers.2]] ([[User talk:Helldivers.2|discuss]] • [[Special:Contributions/Helldivers.2|contribs]]) 17:12, 23 March 2024 (UTC)[[Digital Media Concepts/Helldivers 2 Impact on Gaming Community]] * [[User:Arbalest zy|Arbalest zy]] ([[User talk:Arbalest zy|discuss]] • [[Special:Contributions/Arbalest zy|contribs]]) 03:45, 25 March 2024 (UTC)[[Digital Media Concepts/Proprietary PS Vita memory card]] * [[User:RaymondHuang10|RaymondHuang10]] ([[User talk:RaymondHuang10|discuss]] • [[Special:Contributions/RaymondHuang10|contribs]]) 11:59, 25 March 2024 (UTC)[[Digital Media Concepts/Keshi]] * [[User:Weijian Chen|Weijian Chen]] ([[User talk:Weijian Chen|discuss]] • [[Special:Contributions/Weijian Chen|contribs]]) 21:17, 28 March 2024 (UTC)[[Digital Media Concepts/CPU Evolution]] {{collapse bottom}} === Fall 2023 === {{collapse top|Project List}} * [[User:Homero021004|Homero021004]] ([[User talk:Homero021004|discuss]] • [[Special:Contributions/Homero021004|contribs]]) 06:23, 23 October 2023 (UTC)[[Digital Media Concepts/Autism (ASD) & Social Media]] * -~--([Digital Media Concepts/Transforming Agriculture: A Case Study on Pakistan|)) * [[User:Garylongington|Garylongington]] ([[User talk:Garylongington|discuss]] • [[Special:Contributions/Garylongington|contribs]]) 21:04, 22 October 2023 (UTC)[[Digital Media Concepts/Slender: The Eight Pages]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 16:06, 6 June 2023 (UTC)[[Digital Media Concepts/Wafaa Bilal Example]] * [[User:Archelone|Archelone]] ([[User talk:Archelone|discuss]] • [[Special:Contributions/Archelone|contribs]]) 01:32, 16 October 2023 (UTC)[[Digital Media Concepts/Representation in Baldur's Gate 3]] * [[User:Gauze0001|Gauze0001]] ([[User talk:Gauze0001|discuss]] • [[Special:Contributions/Gauze0001|contribs]]) 01:11, 17 October 2023 (UTC)[[Digital Media Concepts/Cameron's World]] * [[User:Owynmlee|Owynmlee]] ([[User talk:Owynmlee|discuss]] • [[Special:Contributions/Owynmlee|contribs]]) 01:11, 18 October 2023 (UTC)[[Digital Media Concepts/Shrinkwrapping]] * [[User:Atnguyen12|Atnguyen12]] ([[User talk:Atnguyen12|discuss]] • [[Special:Contributions/Atnguyen12|contribs]]) 05:54, 21 October 2023 (UTC)[[Digital Media Concepts/Wonderlab]] * [[User:BinhNguyenSKMT|BinhNguyenSKMT]] ([[User talk:BinhNguyenSKMT|discuss]] * [[Special:Contributions/BinhNguyenSKMT|contribs]]) 01:14, 22 October 2023 (UTC) [[Digital Media Concepts/Education with Generative AI]] * [[User:Jonnmac|Jonnmac]] ([[User talk:Jonnmac|discuss]] • [[Special:Contributions/Jonnmac|contribs]]) 02:50, 23 October 2023 (UTC)[[Digital Media Concepts/Team Fortress 2 Culture]] * [[Digital Media Concepts/Transportation Technologies]] * [[User:Kalvabb|Kalvabb]] ([[User talk:Kalvabb|discuss]] * [[Special:Contributions/Kalvabb|contribs]]) 10:58, 22 October 2023 (UTC) [[Digital Media Concepts/Writing and Artificial Intelligence]] * [[User:Karan 035|Karan 035]] ([[User talk:Karan 035|discuss]] • [[Special:Contributions/Karan 035|contribs]]) 02:07, 23 October 2023 (UTC)[[Digital Media Concepts/Steve jobs and his Leadership styles]] * [[User:Elyssamlee|Elyssamlee]] ([[User talk:Elyssamlee|discuss]] • [[Special:Contributions/Elyssamlee|contribs]]) 02:53, 23 October 2023 (UTC)[[Digital Media Concepts/Steve Lacy]] * [[User:Daniel Lopezzz|Daniel Lopezzz]] ([[User talk:Daniel Lopezzz|discuss]] • [[Special:Contributions/Daniel Lopezzz|contribs]]) 23:40, 19 November 2023 (UTC)[[Digital Media Concepts/Xbox Technological Advances]] {{collapse bottom}} === Spring 2022 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 19:27, 24 February 2022 (UTC) [[Digital Media Concepts/Wafaa Bilal 3]] * [[User:Logain.Abd|Logain.Abd]] ([[User talk:Logain.Abd|discuss]] • [[Special:Contributions/Logain.Abd|contribs]]) 16:55, 1 March 2022 (UTC) [[Digital Media Concepts/Prosthetic Body Parts]] * [[User:Ceruleansnake|Ceruleansnake]] ([[User talk:Ceruleansnake|discuss]] • [[Special:Contributions/Ceruleansnake|contribs]]) 19:29, 24 February 2022 (UTC) [[Digital Media Concepts/ENA (web series)]] * [[User:SPiedyW5|SPiedyW5]] ([[User talk:SPiedyW5|discuss]] • [[Special:Contributions/SPiedyW5|contribs]]) 19:33, 24 February 2022 (UTC) [[Digital Media Concepts/Streaming Platforms]] * [[User:AkhilKandkur|AkhilKandkur]] ([[User talk:AkhilKandkur|discuss]] • [[Special:Contributions/AkhilKandkur|contribs]]) 19:34, 24 February 2022 (UTC) [[Digital Media Concepts/DVD-Video]] * [[User:Adamgmz|Adamgmz]] ([[User talk:Adamgmz|discuss]] • [[Special:Contributions/Adamgmz|contribs]]) 19:47, 24 February 2022 (UTC) [[Digital Media Concepts/Physically Based Rendering]] * [[User:ZhuoL|ZhuoL]] ([[User talk:ZhuoL|discuss]] • [[Special:Contributions/ZhuoL|contribs]]) 20:29, 24 February 2022 (UTC) [[Digital Media Concepts/Audio Editing]] * [[User:WanderingWalruses|WanderingWalruses]] ([[User talk:WanderingWalruses|discuss]] • [[Special:Contributions/WanderingWalruses|contribs]]) 20:32, 24 February 2022 (UTC) [[Digital Media Concepts/Youtube Monetization]] * [[User:Anammughal|Anammughal]] ([[User talk:Anammughal|discuss]] • [[Special:Contributions/Anammughal|contribs]]) 20:36, 24 February 2022 (UTC) [[Digital Media Concepts/Artificial Intelligence in Healthcare]] * [[User:Vlee88|Vlee88]] ([[User talk:Vlee88|discuss]] • [[Special:Contributions/Vlee88|contribs]]) 22:16, 24 February 2022 (UTC) [[Digital Media Concepts/Wacom/digital tablets]] * [[User:TZrng8|TZrng8]] ([[User talk:TZrng8|discuss]] • [[Special:Contributions/TZrng8|contribs]]) 20:07, 25 February 2022 (UTC) [[Digital Media Concepts/Virtual Reality]] * [[User:Nooshas|Nooshas]] ([[User talk:Nooshas|discuss]] • [[Special:Contributions/Nooshas|contribs]]) 04:57, 28 February 2022 (UTC) [[Digital Media Concepts/Eduardo Saverin]] * [[User:Rjoyner1|Rjoyner1]] ([[User talk:Rjoyner1|discuss]] • [[Special:Contributions/Rjoyner1|contribs]]) 19:26, 1 March 2022 (UTC) [[Digital Media Concepts/Higher Learning Podacast]] * [[User:Yaboiversity|Yaboiversity]] ([[User talk:Yaboiversity|discuss]] • [[Special:Contributions/Yaboiversity|contribs]]) 19:36, 1 March 2022 (UTC) [[Digital Media Concepts/Ceramic 3D Printers]] * [[User:Ayeitsemmie|Ayeitsemmie]] ([[User talk:Ayeitsemmie|discuss]] • [[Special:Contributions/Ayeitsemmie|contribs]]) 20:35, 1 March 2022 (UTC) [[Digital Media Concepts/Technology Used In “Rick and Morty”]] * [[User:Navveerbhangal|Navveerbhangal]] ([[User talk:Navveerbhangal|discuss]] • [[Special:Contributions/Navveerbhangal|contribs]]) 19:56, 1 March 2022 (UTC) [[Digital Media Concepts/Lex Fridman]] * [[User:Mayapose|Mayapose]] ([[User talk:Mayapose|discuss]] • [[Special:Contributions/Mayapose|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Hedy Lamarr]] * [[User:Ohlonestudengo12|Ohlonestudengo12]] ([[User talk:Ohlonestudengo12|discuss]] • [[Special:Contributions/Ohlonestudengo12|contribs]]) 00:23, 2 March 2022 (UTC) [[Digital Media Concepts/Apple AirTag]] * [[User:Aayyaz|Aayyaz]] ([[User talk:Aayyaz|discuss]] • [[Special:Contributions/Aayyaz|contribs]]) 20:08, 3 March 2022 (UTC) [[Digital Media Concepts/Speech Recognition]] * [[User:Brana1|Brana1]] ([[User talk:Brana1|discuss]] • [[Special:Contributions/Brana1|contribs]]) 06:56, 9 March 2022 (UTC)[[Digital Media Concepts/Sony Playstation]] {{collapse bottom}} === Fall 2021 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Five Nights at Freddy's]] * [[User:Zkeeler1|Zkeeler1]] ([[User talk:Zkeeler1|discuss]] • [[Special:Contributions/Zkeeler1|contribs]]) 01:28, 22 September 2021 (UTC) [[Digital Media Concepts/The process of Encryption]] * [[User:Ronit Dhir|Ronit Dhir]] ([[User talk:Ronit Dhir|discuss]] • [[Special:Contributions/Ronit Dhir|contribs]]) 07:15, 14 October 2021 (UTC) [[Digital Media Concepts/Shantanu Narayen]] * [[User:Joachanz|Joachanz]] ([[User talk:Joachanz|discuss]] • [[Special:Contributions/Joachanz|contribs]]) 20:32, 28 September 2021 (UTC) [[Digital Media Concepts/Riot Games Gender Discrimination Controversy]] * [[User:SauerPatch|SauerPatch]] ([[User talk:SauerPatch|discuss]] • [[Special:Contributions/SauerPatch|contribs]]) 20:35, 28 September 2021 (UTC) [[Digital Media Concepts/Bitcoin]] * [[User:Vy Trieu|Vy Trieu]] ([[User talk:Vy Trieu|discuss]] • [[Special:Contributions/Vy Trieu|contribs]]) 20:47, 28 September 2021 (UTC) [[Digital Media Concepts/Virtual Influencer]] * [[User:GEEGOLLYY|GEEGOLLYY]] ([[User talk:GEEGOLLYY|discuss]] • [[Special:Contributions/GEEGOLLYY|contribs]]) 21:01, 28 September 2021 (UTC) [[Digital Media Concepts/Valorant]] * [[User:Cheesze|Cheesze]] ([[User talk:Cheesze|discuss]] • [[Special:Contributions/Cheesze|contribs]]) 21:10, 28 September 2021 (UTC) [[Digital Media Concepts/Squid Game S1E6: Gganbu]] * [[User:PcPeou1|PcPeou1]] ([[User talk:PcPeou1|discuss]] • [[Special:Contributions/PcPeou1|contribs]]) 21:19, 28 September 2021 (UTC) [[Digital Media Concepts/Ace Combat 7: Skies Unknown]] * [[User:Ralph Lagmay|Ralph Lagmay]] ([[User talk:Ralph Lagmay|discuss]] • [[Special:Contributions/Ralph Lagmay|contribs]]) 19:24, 3 October 2021 (UTC) [[Digital Media Concepts/Beeple]] * [[User:Nicole Wood|Nicole Wood]] ([[User talk:Nicole Wood|discuss]] • [[Special:Contributions/Nicole Wood]]) 23:29, 5 October 2021 (UTC) [[Digital Media Concepts/Sam Does Arts]] * [[User:Choopa land|Choopa land]] ([[User talk:Choopa land|discuss]] • [[Special:Contributions/Choopa land|contribs]]) 21:31, 7 October 2021 (UTC) [[Digital Media Concepts/Sudha Murty]] * [[User:Alex Walsh|Alex Walsh]] ([[User talk:Alex Walsh|discuss]] • [[Special:Contributions/Alex Walsh|contribs]]) 07:35, 9 October 2021 (UTC) [[Digital Media Concepts/Twitch Hate Raids]] * [[User:Just TMar|Just TMar]] ([[User talk:Just TMar|discuss]] • [[Special:Contributions/Just TMar|contribs]]) 01:54, 11 October 2021 (UTC) [[Digital Media Concepts/Doxing]] * [[User:Ghiyasat|Ghiyasat]] ([[User talk:Ghiyasat|discuss]] • [[Special:Contributions/Ghiyasat|contribs]]) 17:28, 11 October 2021 (UTC) [[Digital Media Concepts/Polygraph]] * [[User:Snuckles49|Snuckles49]] ([[User talk:Snuckles49|discuss]] • [[Special:Contributions/Snuckles49|contribs]]) 00:43, 12 October 2021 (UTC) [[Digital Media Concepts/BattlEye Anti-Cheat]] * [[User:Nino02|Nino02]] ([[User talk:Nino02|discuss]] • [[Special:Contributions/Nino02|contribs]]) (UTC) [[Digital Media Concepts/Video Streaming Services]] * [[User:Ishadamle|Ishadamle]] ([[User talk:Ishadamle|discuss]] • [[Special:Contributions/Ishadamle|contribs]]) 16:56, 12 October 2021 (UTC) [[Digital Media Concepts/Aistė Stancikaitė]] * [[User:Zainab Attarwala|Zainab Attarwala]] ([[User talk:Zainab Attarwala|discuss]] • [[Special:Contributions/Zainab Attarwala|contribs]]) 22:37, 21 September 2021 (UTC) [[Digital Media Concepts/Tokyo Medical University for Rejected Woman]] * [[User:Yusef510|Yusef510]] ([[User talk:Yusef510|discuss]] • [[Special:Contributions/Yusef510|contribs]]) 21:26, 12 October 2021 (UTC) [[Digital Media Concepts/Women In Esports]] * [[User:SLBriscoe|SLBriscoe]] ([[User talk:SLBriscoe|discuss]] • [[Special:Contributions/SLBriscoe|contribs]]) 21:36, 12 October 2021 (UTC) [[Digital Media Concepts/Dark Souls III]] * [[User:Phonemyat238|Phonemyat238]] ([[User talk:Phonemyat238|discuss]] • [[Special:Contributions/Phonemyat238|contribs]]) 23:47, 12 October 2021 (UTC) [[Digital Media Concepts/Dota 2 Esport]] * [[User:JBauer4|JBauer4]] ([[User talk:JBauer4|discuss]] • [[Special:Contributions/JBauer4|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/New World (game)]] * [[User:Rburnham1740|Rburnham1740]] ([[User talk:Rburnham1740|discuss]] • [[Special:Contributions/Rburnham1740|contribs]]) 23:57, 12 October 2021 (UTC) [[Digital Media Concepts/Call Of Duty: Warzone]] * [[User:HellaCinema|HellaCinema]] ([[User talk:HellaCinema|discuss]] • [[Special:Contributions/HellaCinema|contribs]]) 06:39, 13 October 2021 (UTC) [[Digital Media Concepts/New Age Cinema - Film Cinematography]] * [[User:Tiwarisushant|Tiwarisushant]] ([[User talk:Tiwarisushant|discuss]] • [[Special:Contributions/Tiwarisushant|contribs]]) 23:58, 12 October 2021 (UTC) [[Digital Media Concepts/CGI - Uses and Importance]] * [[User:JoseNavarro15|JoseNavarro15]] ([[User talk:JoseNavarro15|discuss]] • [[Special:Contributions/JoseNavarro15|contribs]]) 05:26, 18 October 2021 (UTC) [[Digital Media Concepts/Space X]] {{collapse bottom}} === Fall 2020 === {{collapse top|Project List}} * [[User:Daniyal.ak27|Daniyal.ak27]] ([[User talk:Daniyal.ak27|discuss]] • [[Special:Contributions/Daniyal.ak27|contribs]]) 21:42, 15 October 2020 (UTC) [[/Neuralink/]] * [[User:Alisonmp|Alisonmp]] ([[User talk:Alisonmp|discuss]] • [[Special:Contributions/Alisonmp|contribs]]) 21:41, 15 October 2020 (UTC) [[/Video Streaming Services/]] * [[Special:Contributions/2601:641:400:4F00:39DF:F97A:724A:496D|2601:641:400:4F00:39DF:F97A:724A:496D]] ([[User talk:2601:641:400:4F00:39DF:F97A:724A:496D|discuss]]) 21:40, 15 October 2020 (UTC) [[/Metroidvania/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 03:26, 14 October 2020 (UTC) [[/Zoom Video Communications/]] * [[User:Nidhisgupta|Nidhisgupta]] ([[User talk:Nidhisgupta|discuss]] • [[Special:Contributions/Nidhisgupta|contribs]]) 23:21, 11 October 2020 (UTC) [[/Cyber Crimes/]] * [[User:Melissalop551|Melissalop551]] ([[User talk:Melissalop551|discuss]] • [[Special:Contributions/Melissalop551|contribs]]) 23:40, 9 October 2020 (UTC) [[/Iphone X (Face ID)/]] * [[User:Navink100|Navink100]] ([[User talk:Navink100|discuss]] • [[Special:Contributions/Navink100|contribs]]) 20:03, 8 October 2020 (UTC) [[/The Evolution of Digital Scouting in the NBA/]] * [[User:Rthawani2|Rthawani2]] ([[User talk:Rthawani2|discuss]] • [[Special:Contributions/Rthawani2|contribs]]) 22:35, 6 October 2020 (UTC) [[/LD, 2D, SD, 3D, HD, 4D, 4K, 8K/]] * [[User:Bandz21811|Bandz21811]] ([[User talk:Bandz21811|discuss]] • [[Special:Contributions/Bandz21811|contribs]]) 21:13, 6 October 2020 (UTC) [[/Tesla Inc./]] * [[User:MalakaiCaro|MalakaiCaro]] ([[User talk:MalakaiCaro|discuss]] • [[Special:Contributions/MalakaiCaro|contribs]]) 21:03, 6 October 2020 (UTC) [[/Apple CarPlay/]] * [[User:Jb000ts|Jb000ts]] ([[User talk:Jb000ts|discuss]] • [[Special:Contributions/Jb000ts|contribs]]) 20:46, 6 October 2020 (UTC) [[/Porsche Taycan/]] * [[User:Simplyktp|Simplyktp]] ([[User talk:Simplyktp|discuss]] • [[Special:Contributions/Simplyktp|contribs]]) 00:54, 6 October 2020 (UTC) [[/Animal Crossing: New Horizons/]] * [[User:Ankith Reddy Alwa|Ankith Reddy Alwa]] ([[User talk:Ankith Reddy Alwa|discuss]] • [[Special:Contributions/Ankith Reddy Alwa|contribs]]) 20:14, 1 October 2020 (UTC) [[/Narayana Murthy/]] * [[User:MoisesMorales04|MoisesMorales04]] ([[User talk:MoisesMorales04|discuss]] • [[Special:Contributions/MoisesMorales04|contribs]]) 23:23, 29 September 2020 (UTC) [[/Adaptive Cruise Control/]] * [[User:Rhummdan|Rhummdan]] ([[User talk:Rhummdan|discuss]] • [[Special:Contributions/Rhummdan|contribs]]) 21:45, 29 September 2020 (UTC) [[/NBA 2K/]] * [[User:Yiyi799|Yiyi799]] ([[User talk:Yiyi799|discuss]] • [[Special:Contributions/Yiyi799|contribs]]) 20:36, 29 September 2020 (UTC) [[/Snapchat Lenses/]] * [[User:Oaramirez780|Oaramirez780]] ([[User talk:Oaramirez780|discuss]] • [[Special:Contributions/Oaramirez780|contribs]]) 20:17, 29 September 2020 (UTC) [[/Image Comics/]] * [[User:Coleslamfreckle|Coleslamfreckle]] ([[User talk:Coleslamfreckle|discuss]] • [[Special:Contributions/Coleslamfreckle|contribs]]) 19:36, 23 September 2020 (UTC) [[/Linux Beta/]] * [[User:Shaikhhiba|Shaikhhiba]] ([[User talk:Shaikhhiba|discuss]] • [[Special:Contributions/Shaikhhiba|contribs]]) 02:01, 24 September 2020 (UTC) [[/Cancel Culture/]] * [[User:444juptr|444juptr]] ([[User talk:444juptr|discuss]] • [[Special:Contributions/444juptr|contribs]]) 20:33, 29 September 2020 (UTC) [[/Otaku Life/]] *[[User:Shruthi2001|Shruthi2001]] ([[User talk:Shruthi2001|discuss]] • [[Special:Contributions/Shruthi2001|contribs]]) 20:40, 29 September 2020 (UTC) [[/Social Media Effect on Diet/]] * [[User:Vtran50|Vtran50]] ([[User talk:Vtran50|discuss]] • [[Special:Contributions/Vtran50|contribs]]) 20:51, 29 September 2020 (UTC) [[/Slander/]] * [[User:JayceeLorenzo|JayceeLorenzo]] ([[User talk:JayceeLorenzo|discuss]] • [[Special:Contributions/JayceeLorenzo|contribs]]) 21:24, 29 September 2020 (UTC) [[/Continuous Variable Transmission (CVT)/]] * [[User:IowaneNoka|IowaneNoka]] ([[User talk:IowaneNoka|discuss]] • [[Special:Contributions/IowaneNoka|contribs]]) 03:12, 10 October 2020 (UTC) [[/Call Of Duty:Modern Warfare (2019)/]] * [[User:Kban2001|Kban2001]] ([[User talk:Kban2001|discuss]] • [[Special:Contributions/Kban2001|contribs]]) 16:11, 12 October 2020 (UTC) [[/ARK: Survival Evolved/]] * [[User:Christiand28|Christiand28]] ([[User talk:Christiand28|discuss]] • [[Special:Contributions/Christiand28|contribs]]) 02:21, 13 October 2020 (UTC) [[/Call of Duty:Black Ops 2/]] {{collapse bottom}} === Spring 2020 === {{collapse top|Project List}} * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 22:28, 5 March 2020 (UTC) [[Digital Media Concepts/Postmodern Jukebox]] * [[User:Deuteryium|Deuteryium]] ([[User talk:Deuteryium|discuss]] • [[Special:Contributions/Deuteryium|contribs]]) 23:16, 27 February 2020 (UTC) [[Digital Media Concepts/Breaking the Fourth Wall]] * [[User:PlasticOrPapper|PlasticOrPapper]] ([[User talk:PlasticOrPapper|discuss]] • [[Special:Contributions/PlasticOrPapper|contribs]]) 23:14, 27 February 2020 (UTC) [[Digital Media Concepts/Tom Clancy's Rainbow Six Siege]] * [[User:Miranda Tapioca|Miranda Tapioca]] ([[User talk:Miranda Tapioca|discuss]] • [[Special:Contributions/Miranda Tapioca|contribs]]) 23:04, 25 February 2020 (UTC) [[Digital Media Concepts/Blizzard Entertaiment]] * [[User:Christinuh|Christinuh]] ([[User talk:Christinuh|discuss]] • [[Special:Contributions/Christinuh|contribs]]) 23:07, 25 February 2020 (UTC) [[Digital Media Concepts/Breaking News on Twitter]] * [[User:Emartinez40|Emartinez40]] ([[User talk:Emartinez40|discuss]] • [[Special:Contributions/Emartinez40|contribs]]) 23:09, 25 February 2020 (UTC) [[Digital Media Concepts/The Witcher (Game Series)]] * [[User:MarkxG|MarkxG]] ([[User talk:MarkxG|discuss]] • [[Special:Contributions/MarkxG|contribs]]) 23:26, 25 February 2020 (UTC) [[Digital Media Concepts/Reddit]] * [[User:Dmusselwhite1|Dmusselwhite1]] ([[User talk:Dmusselwhite1|discuss]] • [[Special:Contributions/Dmusselwhite1|contribs]]) 23:15, 25 February 2020 (UTC) [[Digital Media Concepts/CGI]] * [[User:Ericahy|Ericahy]] ([[User talk:Ericahy|discuss]] • [[Special:Contributions/Ericahy|contribs]]) 23:17, 25 February 2020 (UTC) [[Digital Media Concepts/Siri]] * [[User:Aphon17|Aphon17]] ([[User talk:Aphon17|discuss]] • [[Special:Contributions/Aphon17|contribs]]) 23:23, 25 February 2020 (UTC) [[Digital Media Concepts/Inside (video game)]] * [[User:Mpaing2|Mpaing2]] ([[User talk:Mpaing2|discuss]] • [[Special:Contributions/Mpaing2|contribs]]) 22:15, 27 February 2020 (UTC) [[Digital Media Concepts/Apple Pencil]] * [[User:TakeCare2021|TakeCare2021]] ([[User talk:TakeCare2021|discuss]] • [[Special:Contributions/TakeCare2021|contribs]]) 22:52, 27 February 2020 (UTC) [[Digital Media Concepts/Take Care (Album)]] * [[User:DixonCider1331|DixonCider1331]] ([[User talk:DixonCider1331|discuss]] • [[Special:Contributions/DixonCider1331|contribs]]) 22:55, 27 February 2020 (UTC) [[Digital Media Concepts/Riot Games]] * [[User:John.rod22|John.rod22]] ([[User talk:John.rod22|discuss]] • [[Special:Contributions/John.rod22|contribs]]) 23:01, 27 February 2020 (UTC) [[Digital Media Concepts/Grand Theft Auto V]] * [[User:Ankits2001|Ankits2001]] ([[User talk:Ankits2001|discuss]] • [[Special:Contributions/Ankits2001|contribs]]) 23:06, 27 February 2020 (UTC) [[Digital Media Concepts/iPhone 4s (Siri)]] * [[User:MichaelP75075|MichaelP75075]] ([[User talk:MichaelP75075|discuss]] • [[Special:Contributions/MichaelP75075|contribs]]) 22:10, 5 March 2020 (UTC) [[Digital Media Concepts/Black Mesa]] * [[User:Kfernes1|Kfernes1]] ([[User talk:Kfernes1|discuss]] • [[Special:Contributions/Kfernes1|contribs]]) 22:38, 5 March 2020 (UTC) [[Digital Media Concepts/Joe Rogan Experience]] * [[Special:Contributions/207.62.190.33|207.62.190.33]] ([[User talk:207.62.190.33|discuss]]) 23:42, 5 March 2020 (UTC) [[Digital Media Concepts/Tinder]] * [[User:Atul.mav|Atul.mav]] ([[User talk:Atul.mav|discuss]] • [[Special:Contributions/Atul.mav|contribs]]) 17:48, 7 March 2020 (UTC) [[Digital Media Concepts/Ralph Lauren]] * [[User:Namesshaikh|Namesshaikh]] ([[User talk:Namesshaikh|discuss]] • [[Special:Contributions/Namesshaikh|contribs]]) 22:19, 9 March 2020 (UTC) [[Digital Media Concepts/CD Projekt Red]] {{collapse bottom}} === Fall 2019 === {{collapse top|Project List}} * [[User:HPatel40|HPatel40]] ([[User talk:HPatel40|discuss]] • [[Special:Contributions/HPatel40|contribs]]) 19:23, 17 September 2019 (UTC) [[Digital Media Concepts/5G Technologies]] * [[User:Hpatel42|Hpatel42]] ([[User talk:Hpatel42|discuss]] • [[Special:Contributions/Hpatel42|contribs]]) 19:01, 26 September 2019 (UTC)[[Digital Media Concepts/Larry Page]] * [[User:CRichards17|CRichards17]] ([[User talk:CRichards17|discuss]] • [[Special:Contributions/CRichards17|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/White Supremacy on the Internet]] * [[User:Waterbottle50|Waterbottle50]] ([[User talk:Waterbottle50|discuss]] • [[Special:Contributions/Waterbottle50|contribs]]) 19:55, 26 September 2019 (UTC)[[Digital Media Concepts/Color-blind lens/EnChroma]] * [[User:Mcaya00|Mcaya00]] ([[User talk:Mcaya00|discuss]] • [[Special:Contributions/Mcaya00|contribs]]) 19:49, 17 September 2019 (UTC) [[Digital Media Concepts/Markus Perrson]] * [[User:Qinyu Chen|Qinyu Chen]] ([[User talk:Qinyu Chen|discuss]] • [[Special:Contributions/Qinyu Chen|contribs]]) 19:36, 17 September 2019 (UTC) [[Digital Media Concepts/teamLab]] * [[User:Soyoboy.exe|Soyboy.exe]] ([[User talk:Soyboy.exe|discuss]] • [[Special:Contributions/Soyboy.exe|contribs]]) 20:04, 17 September 2019 (UTC) [[Digital Media Concepts/Nintendo Switch]] * [[User:Yjiang26|Yjiang26]] ([[User talk:Yjiang26|discuss]] • [[Special:Contributions/Yjiang26|contribs]]) 19:38, 17 September 2019 (UTC) [[Digital Media Concepts/Alipay]] * [[User:Lmurffrenninger1|Lmurffrenninger1]] ([[User talk:Lmurffrenninger1|discuss]] • [[Special:Contributions/Lmurffrenninger1|contribs]]) 19:39, 17 September 2019 (UTC) [[Digital Media Concepts/MagicBands at Disneyland]] * [[User:Jalvarellos1|Jalvarellos1]] ([[User talk:Jalvarellos1|discuss]] • [[Special:Contributions/Jalvarellos1|contribs]]) 19:41, 17 September 2019 (UTC) [[Digital Media Concepts/Rocket League]] * [[Adam3318|Adam3318]] ([[User talk:Adam3318|discuss]] • [[Special:Contributions/Adam3318|contribs]]) 19:46, 17 September 2019 (UTC) [[Digital Media Concepts/3D scanning with drones]] * [[User:SanIam|SanIam]] ([[User talk:SanIam|discuss]] • [[Special:Contributions/SanIam|contribs]]) 19:57, 17 September 2019 (UTC) [[Digital Media Concepts/The Legend Of Zelda: Breath of the Wild]] * [[User:Nphan16|Nphan16]] ([[User talk:Nphan16|discuss]] • [[Special:Contributions/Nphan16|contribs]]) 20:02, 17 September 2019 (UTC) [[Digital Media Concepts/Hannah Alexander]] * [[User:Vluong6|Vluong6]] ([[User talk:Vluong6|discuss]] • [[Special:Contributions/Vluong6|contribs]]) 20:24, 17 September 2019 (UTC) [[Digital Media Concepts/Cyberware]] * [[User:Philventi|Philventi]] ([[User talk:Philventi|discuss]] • [[Special:Contributions/Philventi|contribs]]) 22:33, 17 September 2019 (UTC) [[Digital Media Concepts/iPadOS]] * [[User:Jwendland1|Jwendland1]] ([[User talk:Jwendland1|discuss]] • [[Special:Contributions/Jwendland1|contribs]]) 19:05, 19 September 2019 (UTC)[[Digital Media Concepts/Infinadeck Omnidirectional Treadmill]] * [[User:Ranle7|Ranle7]] ([[User talk:Ranle7|discuss]] • [[Special:Contributions/Ranle7|contribs]]) 19:06, 26 September 2019 (UTC)[[Digital Media Concepts/osu!]] * [[User:Andrewyamasaki|Andrewyamasaki]] ([[User talk:Andrewyamasaki|discuss]] • [[Special:Contributions/Andrewyamasaki|contribs]]) 19:48, 17 September 2019 (UTC) [[Digital Media Concepts/Competitive Super Smash Bros Ultimate]] * [[User:Samuel3571|Samuel3571]] ([[User talk:Samuel3571|discuss]] • [[Special:Contributions/Samuel3571|contribs]]) 20:35, 19 September 2019 (UTC)[[Digital Media Concepts/Forensic Technology]] * [[User:Lmedina19|Lmedina19]] ([[User talk:Lmedina19|discuss]] • [[Special:Contributions/Lmedina19|contribs]]) 20:00, 26 September 2019 (UTC)[[Digital Media Concepts/Hayley Kiyoko]] * [[User:Djh42|Djh42]] ([[User talk:Djh42|discuss]] • [[Special:Contributions/Djh42|contribs]]) 20:25, 26 September 2019 (UTC)[[Digital Media Concepts/Nintendo Switch Controller]] * [[User:CharlieDub|CharlieDub]] ([[User talk:CharlieDub|discuss]] • [[Special:Contributions/CharlieDub|contribs]]) 19:42, 17 September 2019 (UTC) [[Digital Media Concepts/Porsche Ignition]] * [[User:Erick.Flore|Erick.Flore]] ([[User talk:Erick.Flore|discuss]] • [[Special:Contributions/Erick.Flore|contribs]]) 20:40, 26 September 2019 (UTC)[[Digital Media Concepts/The video hosting site BitChute]] * [[User:Dbarquin1|Dbarquin1]] ([[User talk:Dbarquin1|discuss]] • [[Special:Contributions/Dbarquin1|contribs]]) 17:28, 27 September 2019 (UTC)[[Digital Media Concepts/Custom Built Computer]] * [[User:Spcharc|Spcharc]] ([[User talk:Spcharc|discuss]] • [[Special:Contributions/Spcharc|contribs]]) 18:07, 29 September 2019 (UTC) [[Digital Media Concepts/RSA (cryptosystem)]] * [[User:Krishnan1000|Krishnan1000]] ([[User talk:Krishnan1000|discuss]] • [[Special:Contributions/Krishnan1000|contribs]]) 18:36, 29 September 2019 (UTC) [[Digital Media Concepts/Competitive Pokemon Battling]] {{collapse bottom}} === Spring 2019 === {{collapse top|Project List}} * [[User:Mknight77|Mknight77]] ([[User talk:Mknight77|discuss]] • [[Special:Contributions/Mknight77|contribs]]) 22:52, 26 February 2019 (UTC) [[Digital Media Concepts/Mayhem fest]] * [[User:Samirjan0420|Samirjan0420]] ([[User talk:Samirjan0420|discuss]] • [[Special:Contributions/Samirjan0420|contribs]]) 22:05, 21 February 2019 (UTC) [[/The most hated car/]] * [[User:Tobilansangan|Tobilansangan]] ([[User talk:Tobilansangan|discuss]] • [[Special:Contributions/Tobilansangan|contribs]]) 23:39, 19 February 2019 (UTC) [[/My Beautiful Dark Twisted Fantasy/]] * [[User:alyssamunoz|alyssamunoz]] ([[User talk: alyssamunoz|discuss]] • [[Special:Contributions/alyssamunoz|contribs]]) 14:57, 19 February 2019 (UTC) [[/Dr.Pepper/]] * [[User:omergreengiant|omergreengiant]] ([[User talk:omergreengiant|discuss]] • [[Special:Contributions/omergreengiant|contribs]]) 14:57, 19 February 2019 (UTC) [[/Kabul, Afghanistan/]] * [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 22:14, 19 February 2019 (UTC) [[/Wafaa Bilal/]] * [[User:Gum-POP|Gum-POP]] ([[User talk:Gum-POP|discuss]] • [[Special:Contributions/Gum-POP|contribs]]) 22:37, 14 February 2019 (UTC) [[/Gaming Desktop VS Gaming Laptop/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 22:42, 14 February 2019 (UTC) [[/Virtual Self/]] * [[User:Tvo36|Tvo36]] ([[User talk:Tvo36|discuss]] • [[Special:Contributions/Tvo36|contribs]]) 23:13, 14 February 2019 (UTC) [[/Markelle Fultz/]] * [[User:iluca|iluca]] ([[User talk:iluca|discuss]] • [[Special:Contributions/iluca|contribs]]) 23:14, 14 February 2019 (UTC) [[/Nagishiro Mito/]] * [[User:Justinchau|Justinchau]] ([[User talk:Justinchau|discuss]] • [[Special:Contributions/Justinchau|contribs]]) 23:16, 14 February 2019 (UTC) [[/Best Pizza/]] * [[User:Ashwinjambu15|Ashwinjambu15]] ([[User talk:Ashwinjambu15|discuss]] • [[Special:Contributions/Ashwinjambu15|contribs]]) 23:19, 14 February 2019 (UTC) [[/Mos Def & Talib Kweli Are Black Star/]] * [[User:Ationgson|Ationgson]] ([[User talk:Ationgson|discuss]] • [[Special:Contributions/Ationgson|contribs]]) 23:24, 14 February 2019 (UTC) [[/2JZ Engine/]] * [[User:Ddimitrov1|Ddimitrov1]] ([[User talk:Ddimitrov1|discuss]] • [[Special:Contributions/Ddimitrov1|contribs]]) 23:29, 14 February 2019 (UTC) [[/Theatre Of Blood/]] * [[User:Gli15|Gli15]] ([[User talk:Gli15|discuss]] • [[Special:Contributions/Gli15|contribs]]) 22:12, 19 February 2019 (UTC) [[/Matcha/]] * [[User:Akanijade|Akanijade]] ([[User talk:Akanijade|discuss]] • [[Special:Contributions/Akanijade|contribs]]) 22:39, 19 February 2019 (UTC) [[/Batik/]] * [[User:BrokenConsole|BrokenConsole]] ([[User talk:BrokenConsole|discuss]] • [[Special:Contributions/BrokenConsole|contribs]]) 22:46, 19 February 2019 (UTC) [[/Custom Built Gaming Computers/]] * [[User:Lord Arugula|Lord Arugula]] ([[User talk:Lord Arugula|discuss]] • [[Special:Contributions/Lord Arugula|contribs]]) 22:47, 19 February 2019 (UTC) [[/Aurelio Voltaire/]] * [[User:Danialmirza99|Danialmirza99]] ([[User talk:Danialmirza99|discuss]] • [[Special:Contributions/Danialmirza99|contribs]]) 22:50, 19 February 2019 (UTC) [[/Spelling Bee (1938)/]] * [[User:Janine8100|Janine8100]] ([[User talk:Janine8100|discuss]] • [[Special:Contributions/Janine8100|contribs]]) 22:52, 19 February 2019 (UTC) [[/Brunei Rainforests/]] * [[User:D_Sh1nra|D_Sh1nra]] ([[User talk:D_Sh1nra|discuss]] • [[Special:Contributions/D_Sh1nra|contribs]]) 15:07, 19 February 2019 (UTC) [[/Phosphenes/]] * [[User:Jameslacdao|Jameslacdao]] ([[User talk:Jameslacdao|discuss]] • [[Special:Contributions/Jameslacdao|contribs]]) 23:34, 19 February 2019 (UTC) [[/Love of my Life (Queen song)/]] * [[User:Janinaengo|Janinaengo]] ([[User talk:Janinaengo|discuss]] • [[Special:Contributions/Janinaengo|contribs]]) 06:04, 20 February 2019 (UTC) [[/Influence of 90s sitcom Friends/]] * [[User:Mdapostol|Mdapostol]] ([[User talk:Mdapostol|discuss]] • [[Special:Contributions/Mdapostol|contribs]]) 01:32, 26 February 2019 (UTC) [[/Regions of San Jose, California/]] * [[User:HellBlossoms|HellBlossoms]] ([[User talk:HellBlossoms|discuss]] • [[Special:Contributions/HellBlossoms|contribs]]) 14:13, 26 February 2019 (UTC) [[/Bill Kaulitz/]] * [[User:Nintenchris5963|Nintenchris5963]] ([[User talk:Nintenchris5963|discuss]] • [[Special:Contributions/Nintenchris5963|contribs]]) 22:28, 28 February 2019 (UTC) [[/CrossCode/]] {{collapse bottom}} === Fall 2018 === {{collapse top|Project List}} * [[User:J3tR0cK3t|J3tR0cK3t]] ([[User talk:J3tR0cK3t|discuss]] • [[Special:Contributions/J3tR0cK3t|contribs]]) 19:55, 13 September 2018 (UTC) [[/Streets of Rage (Axel Stone)/]] * [[User:William Leber|William Leber]] ([[User talk:William Leber|discuss]] • [[Special:Contributions/William Leber|contribs]]) 19:56, 13 September 2018 (UTC) [[/Ethoslab/]] * [[User:Vicpimentel99|Vicpimentel99]] ([[User talk:Vicpimentel99|discuss]] • [[Special:Contributions/Vicpimentel99|contribs]]) 20:19, 13 September 2018 (UTC) [[/Micah Mahinay/]] * [[User:Magtotodile|Magtotodile]] ([[User talk:Magtotodile|discuss]] • [[Special:Contributions/Magtotodile|contribs]]) 20:21, 13 September 2018 (UTC) [[/Shelter (Song by Porter Robinson & Madeon)/]] * [[User:Ahuq2|Ahuq2]] ([[User talk:Ahuq2|discuss]] • [[Special:Contributions/Ahuq2|contribs]]) 20:25, 13 September 2018 (UTC) [[/Yakuza (video game series)/]] * [[User:Plam153|Plam153]] ([[User talk:Plam153|discuss]] • [[Special:Contributions/Plam153|contribs]]) 20:22, 13 September 2018 (UTC) [[/Trinh Cong Son (Vietnamese songwriter)/]] * [[User:DoomFry68|DoomFry68]] ([[User talk:DoomFry68|discuss]] • [[Special:Contributions/DoomFry68|contribs]]) 20:23, 13 September 2018 (UTC) [[/Yakuza: Goro Majima/]] * [[User:FizzahKafil|FizzahKafil]] ([[User talk:FizzahKafil|discuss]] • [[Special:Contributions/FizzahKafil|contribs]]) 20:24, 13 September 2018 (UTC) [[/Little Big Planet/]] * [[User:Dursa1|Dursa S]] ([[User talk:Dursa1|discuss]] • [[Special:Contributions/Dursa1|contribs]]) 20:24, 13 September 2018 (UTC) [[/r.h. Sin/]] * [[User:ZaroonIqbal|ZaroonIqbal]] ([[User talk:ZaroonIqbal|discuss]] • [[Special:Contributions/ZaroonIqbal|contribs]]) 20:25, 13 September 2018 (UTC) [[/Fortnite/]] * [[User:Lmascardo1|Lmascardo1]] ([[User talk:Lmascardo1|discuss]] • [[Special:Contributions/Lmascardo1|contribs]]) 20:31, 13 September 2018 (UTC) [[/Stardew Valley/]] * [[User:Aungthuhtet245|Aungthuhtet245]] ([[User talk:Aungthuhtet245|discuss]] • [[Special:Contributions/Aungthuhtet245|contribs]]) 20:32, 13 September 2018 (UTC) [[/Science Behind Pixar/]] * [[User:Whill4|Whill4]] ([[User talk:Whill4|discuss]] • [[Special:Contributions/Whill4|contribs]]) 15:06, 17 September 2018 (UTC) [[ /Mariner's Apartment Complex (Lana Del Rey song)/]] * [[User:WontonsofDMG|WontonsofDMG]] ([[User talk:WontonsofDMG|discuss]] • [[Special:Contributions/WontonsofDMG|contribs]]) 19:03, 18 September 2018 (UTC) [[/Heroes of the Storm/]] * [[User:PATISBACK9|PATISBACK9]] ([[User talk:PATISBACK9|discuss]] • [[Special:Contributions/PATISBACK9|contribs]]) 19:19, 18 September 2018 (UTC) [[/DJ screw/]] * [[User:EnderClevToby|EnderClevToby]] ([[User talk:EnderClevToby|discuss]] • [[Special:Contributions/EnderClevToby|contribs]]) 12:15, 18 September 2018 (UTC) [[/Matthew Fredrick/]] * [[User:Danny M487|Danny M487]] ([[User talk:Danny M487|discuss]] • [[Special:Contributions/Danny M487|contribs]]) 19:30, 18 September 2018 (UTC) [[/Casey Neistat/]] * [[User:Itsjustkulsoom|Itsjustkulsoom]] ([[User talk:Itsjustkulsoom|discuss]] • [[Special:Contributions/Itsjustkulsoom|contribs]]) 19:31, 18 September 2018 (UTC) [[/Zedd/]] * [[User:Josemazon123|Josemazon123]] ([[User talk:Josemazon123|discuss]] • [[Special:Contributions/Josemazon123|contribs]]) 20:06, 18 September 2018 (UTC) [[/BROCKHAMPTON/]] * [[User:Ahernandez85|Ahernandez85]] ([[User talk:Ahernandez85|discuss]] • [[Special:Contributions/Ahernandez85|contribs]]) 19:17, 20 September 2018 (UTC) [[/Legend Of Zelda: Ocarina of Time/]] * [[User:JackyLu01|JackyLu01]] ([[User talk:JackyLu01|discuss]] • [[Special:Contributions/JackyLu01|contribs]]) 12:30, 20 September 2018 (UTC) [[/Smartisan/]] * [[User:Manohar singh 32|Manohar singh 32]] ([[User talk:Manohar singh 32|discuss]] • [[Special:Contributions/Manohar singh 32|contribs]]) 19:51, 20 September 2018 (UTC) [[/Motion Capture/]] * [[User:Zwo1|Zwo1]] ([[User talk:Zwo1|discuss]] • [[Special:Contributions/Zwo1|contribs]]) 23:57, 23 September 2018 (UTC) [[/DJI Mavic 2 Pro/]] * [[User:Hartveit|Hartveit]] ([[User talk:Hartveit|discuss]] • [[Special:Contributions/Hartveit|contribs]]) 01:49, 25 September 2018 (UTC) [[/Hurdal Verk Folk High School/]] * [[User:p_ng_r|p_ng_r]] ([[User talk:p_ng_r|discuss]] • [[Special:Contributions/p_ng_r|contribs]]) 12:41 25 September 2018 (UTC) [[/Wenqing Yan/]] {{collapse bottom}} === Spring 2018 === {{collapse top|Project List}} * [[User:Ireicher2|Ireicher2]] 01:01, 15 August 2016 (UTC) [[/Wafaa Bilal/]] * [[User:Wompaku|Wompaku]] ([[User talk:Wompaku|discuss]] • [[Special:Contributions/Wompaku|contribs]]) 22:21, 13 February 2018 (UTC) [[/Life is Strange/]] * [[User:Hamitaro|Hamitaro]] ([[User talk:Hamitaro|discuss]] • [[Special:Contributions/Hamitaro|contribs]]) 22:22, 13 February 2018 (UTC) [[/Persona 5/]] * [[User:Shayan223|Sean223]] ([[User talk:Shayan223|discuss]] • [[Special:Contributions/Shayan223|contribs]]) 22:33, 13 February 2018 (UTC) [[/David Patrick Crane/]] * [[User:Touch92|Touch92]] ([[User talk:Touch92|discuss]] • [[Special:Contributions/Touch92|contribs]]) 22:29, 13 February 2018 (UTC) [[/3D Printing/]] * [[User:Peg123|Peg123]] ([[User talk:Peg123|discuss]] • [[Special:Contributions/Peg123|contribs]]) 22:32, 13 February 2018 (UTC) [[/Dota 2/]] * [[User:Shadow1942oohwa|Shadow1942oohwa]] ([[User talk:Shadow1942oohwa|discuss]] • [[Special:Contributions/Shadow1942oohwa|contribs]]) 22:34, 13 February 2018 (UTC) [[/Dodge Charger Daytona/]] * [[User:Skullallen|Skullallen]] ([[User talk:Skullallen|discuss]] • [[Special:Contributions/Skullallen|contribs]]) 22:38, 13 February 2018 (UTC) [[/DJI/]] * [[User:Johnbcuong|Johnbcuong]] ([[User talk:Johnbcuong|discuss]] • [[Special:Contributions/Johnbcuong|contribs]]) 22:39, 13 February 2018 (UTC) [[/Nguyen Tu Quang/]] * [[User:W popal1|W popal1]] ([[User talk:W popal1|discuss]] • [[Special:Contributions/W popal1|contribs]]) 22:36, 15 February 2018 (UTC) [[/Lilly Singh/]] * [[User:Sankeerth1017|Sankeerth1017]] ([[User talk:Sankeerth1017|discuss]] • [[Special:Contributions/Sankeerth1017|contribs]]) 22:41, 13 February 2018 (UTC) [[/Pewdiepie/]] * [[User:Jpierceall1|Jpierceall1]] ([[User talk:Jpierceall1|discuss]] • [[Special:Contributions/Jpierceall1|contribs]]) 22:49, 13 February 2018 (UTC) [[/The World of FPS/]] *[[User:ShardolBGupta|ShardolBGupta]] ([[User talk:ShardolBGupta|discuss]] • [[Special:Contributions/ShardolBGupta|contribs]]) 22:42, 13 February 2018 (UTC) [[/Chevrolet Volt/]] * [[User:JoshM75401|JoshM75401]] ([[User talk:JoshM75401|discuss]] • [[Special:Contributions/JoshM75401|contribs]]) 22:49, 13 February 2018 (UTC) [[/Jack Thompson/]] * [[User:Thaliatorres|Thaliatorres]] ([[User talk:Thaliatorres|discuss]] • [[Special:Contributions/Thaliatorres|contribs]]) 23:04, 15 February 2018 (UTC) [[/Laserphaco Probe/]] * [[User:YuuchyHongroy|YuuchyHongroy]] ([[User talk:YuuchyHongroy|discuss]] • [[Special:Contributions/YuuchyHongroy|contribs]]) 23:23, 15 February 2018 (UTC) [[/Photoshop/]] * [[User:stzyjpg|stzyjpg]] ([[User talk:stzyjpg|discuss]] • [[Special:Contributions/stzyjpg|contribs]]) 23:23, 18 February 2018 (UTC) [[/Jeff Bezos/]] {{collapse bottom}} === Fall 2017 === {{collapse top|Project List}} *[[User:Kbearrr|Kbearrr]] ([[User talk:Kbearrr|discuss]] • [[Special:Contributions/Kbearrr|contribs]]) 19:20, 19 September 2017 (UTC) [[/Detriments of the Internet/]] *[[User:Ratalouie|Ratalouie]] ([[User talk:Ratalouie|discuss]] • [[Special:Contributions/Ratalouie|contribs]]) 19:45, 19 September 2017 (UTC) [[/Overwatch/]] *[[User:Inadversity574|Inadversity574]] ([[User talk:Inadversity574|discuss]] • [[Special:Contributions/Inadversity574|contribs]]) 19:44, 19 September 2017 (UTC) [[/Fire Emblem/]] *[[User:Dwingdwang|Dwingdwang]] ([[User talk:Dwingdwang|discuss]] • [[Special:Contributions/Dwingdwang|contribs]]) 20:02, 19 September 2017 (UTC) [[/Fixed Gear Bikes/]] * [[User:Isaacpacheco12|Isaacpacheco12]] ([[User talk:Isaacpacheco12|discuss]] • [[Special:Contributions/Isaacpacheco12|contribs]]) 20:12, 19 September 2017 (UTC) [[/Amazon Echo/]] * [[User:Gdonpadlan|Gdonpadlan]] ([[User talk:Gdonpadlan|discuss]] • [[Special:Contributions/Gdonpadlan|contribs]]) 20:21, 19 September 2017 (UTC) [[/Nikon D850/]] *[[User:Cdb1015|Cdb1015]] ([[User talk:Cdb1015|discuss]] • [[Special:Contributions/Cdb1015|contribs]]) 21:59, 19 September 2017 (UTC) [[/Visual Effects in Television and Movies/]] * [[User:Sjsanchez|Sjsanchez]] ([[User talk:Sjsanchez|discuss]] • [[Special:Contributions/Sjsanchez|contribs]]) 01:11, 20 September 2017 (UTC) [[/Ryan Higa/]] * [[User:Aevans28|Aevans28]] ([[User talk:Aevans28|discuss]] • [[Special:Contributions/Aevans28|contribs]]) 03:56, 20 September 2017 (UTC) [[/Patty Jenkins/]] * [[User:Johnlee1234594|Johnlee1234594]] ([[User talk:Johnlee1234594|discuss]] • [[Special:Contributions/Johnlee1234594|contribs]]) 11:17, 19 September 2017 (UTC) [[/Tom Hanks/]] * [[User:Michaelellasos|Michaelellasos]] ([[User talk:Michaelellasos|discuss]] • [[Special:Contributions/Michaelellasos|contribs]]) 21:36, 20 September 2017 (UTC) [[/Counter Strike: Global Offensive/]] * [[bilal621]] ([[User talk:bilal621|discuss]] • [[Special:Contributions/bilal621|contribs]]) 22:03, 20 September 2017 (UTC) [[/Call of Duty: Modern Warfare 2/]] * [[User:Alscylla|Alscylla]] ([[User talk:Alscylla|discuss]] • [[Special:Contributions/Alscylla|contribs]]) 13:11, 21 September 2017 (UTC) [[/Final Fantasy Record Keeper/]] * [[User:Kshadd15|Kshadd15]] ([[User talk:Kshadd15|discuss]] • [[Special:Contributions/Kshadd15|contribs]]) 10:59, 21 September 2017 (UTC) [[/Artificial Intelligence/]] * [[User:Greentea456|Greentea456]] ([[User talk:Greentea456|discuss]] • [[Special:Contributions/Greentea456|contribs]]) 18:29, 21 September 2017 (UTC) [[Digital Media Concepts/Team Fortress 2|Team Fortress 2]] * [[User:Mehraan01|Mehraan01]] ([[User talk:Mehraan01|discuss]] • [[Special:Contributions/Mehraan01|contribs]]) 19:43, 21 September 2017 (UTC) [[/Tesla, Inc./]] * [[User:Zblazin|Zblazin]] ([[User talk:Zblazin|discuss]] • [[Special:Contributions/Zblazin|contribs]]) 20:12, 19 September 2017 (UTC) [[/Graphics processing unit/]] {{collapse bottom}} === Spring 2017 === {{collapse top|Project List}} *[[User:Mattvs1|Mattvs1]] ([[User talk:Mattvs1|discuss]] • [[Special:Contributions/Mattvs1|contribs]]) 22:56, 7 February 2017 (UTC) [[/No Man's Sky/]] * [[User:Ebondoc98|Ebondoc98]] ([[User talk:Ebondoc98|discuss]] • [[Special:Contributions/Ebondoc98|contribs]]) 22:26, 9 February 2017 (UTC) [[/League of Legends/]] * [[User:Dmendoza1|Dmendoza1]] ([[User talk:Dmendoza1|discuss]] • [[Special:Contributions/Dmendoza1|contribs]]) 22:44, 9 February 2017 (UTC) [[/The Legend Of Zelda/]] * [[User:Mmmarrufo|Mmmarrufo]] ([[User talk:Mmmarrufo|discuss]] • [[Special:Contributions/Mmmarrufo|contribs]]) 22:58, 9 February 2017 (UTC) [[/Gabe Newell/]] * [[User:Jfriedenberg|Jfriedenberg]] ([[User talk:Jfriedenberg|discuss]] • [[Special:Contributions/Jfriedenberg|contribs]]) 23:19, 9 February 2017 (UTC) [[/Shinji Mikami/]] * [[User:AKW2017|AKW2017]] ([[User talk:AKW2017|discuss]] • [[Special:Contributions/AKW2017|contribs]]) 23:32, 9 February 2017 (UTC) [[/Wii U/]] * [[User:Wramos1|Wramos1]] ([[User talk:Wramos1|discuss]] • [[Special:Contributions/Wramos1|contribs]]) 23:46, 9 February 2017 (UTC) [[/John Lasseter/]] * [[User:Yjohar1|Yjohar1]] ([[User talk:Yjohar1|discuss]] • [[Special:Contributions/Yjohar1|contribs]]) 23:48, 9 February 2017 (UTC) [[/Video Game Aggression/]] * [[User:Dzhang17|Dzhang17]] ([[User talk:Dzhang17|discuss]] • [[Special:Contributions/Dzhang17|contribs]]) 23:51, 9 February 2017 (UTC) [[/Fan Bingbing/]] * [[User:Dougalhathaway|Dougalhathaway]] ([[User talk:Dougalhathaway|discuss]] • [[Special:Contributions/Dougalhathaway|contribs]]) 00:04, 10 February 2017 (UTC) [[/Rayman/]] * [[User:Mochill|Mochill]] ([[User talk:Mochill|discuss]] • [[Special:Contributions/Mochill|contribs]]) 04:05, 10 February 2017 (UTC) [[/Commercial Head-up displays/]] *[[User:Hdo18|Hdo18]] ([[User talk:Hdo18|discuss]] • [[Special:Contributions/Hdo18|contribs]]) 18:38, 13 February 2017 (UTC) [[/Cory Arcangel/]] * [[User:GnaF69|GnaF69]] ([[User talk:GnaF69|discuss)]] • [[Special:Contributions/GnaF69|contribs]]) 20:22, 13 February 2017 (UTC) [[/John Romero/]] * [[User:Blizz23|Blizz23]] ([[User talk:Blizz23|discuss]] • [[Special:Contributions/Blizz23|contribs]]) 23:45, 14 February 2017 (UTC) [[/Archaeology Robots/]] * [[User:Bchu55|Bchu55]] ([[User talk:Bchu55|discuss]] • [[Special:Contributions/Bchu55|contribs]]) 23:50, 14 February 2017 (UTC) [[/Mark Zuckerberg/]] {{collapse bottom}} '''4. Research your topic'''<br> Look for at least three kinds of sources: # Articles, any published material, interviews, videos and information you can use for your own contributions to the site # Internet sources that can be linked to your page including multimedia content (please make sure that images and other visual materials not restricted by copyright)<br><br> '''5. Add or edit headings such as:''' * Early Life and Education or Biography * Career * Legacy * Personal Life * Exhibitions * Awards and Nominations (if relevant) * Bibliography or Further Reading * History * Software or Hardware * References * External Links For a detailed editing reference, go to: [[Wikiversity:FAQ/Editing|Wikiversity Editing Reference]] <br><br> '''6. Need Images?''' You can find images that are licensed under the Creative Comments license here [https://commons.wikimedia.org/wiki/Main_Page/ Wikimedia Commons]<br><br> Include <nowiki>[[Category:Digital Media Concepts]]</nowiki> at the bottom of the page to move the page into the assignment category. == Citations == APA style <ref>"Name of Article," accessed March 11, 2015, http://www.someURL.com.</ref> Chicago Style: <ref>Name of Article (for a source that does not list a copyright date, you use (n.d.) ). Retrieved October 15, 2015, from  http://www.someURL.com</ref> == Example of an annotated image == [[File:Great Wall of China in 2014.jpg|thumb|Great Wall of China in 2014]] <br style="clear: both;"> == Example of an [[Template:Infobox|infobox]] == {{Infobox |name = Wafaa Bilal |image = [[File:Wafaa-bilal2.JPG|thumb]] |title = Artist Wafaa Bilal |headerstyle = background:#ccf; |labelstyle = background:#ddf; |label1 = |data1 = Wafaa Bilal reading at Lannan Center for Poetics and Social Practice, Georgetown University |label2 = Nationality |data2 = {{{item_one| Iraqi American}}} |label3 = Born |data3 = {{{item_two|June 10, 1966}}} |label4 = Field of Research |data4 = {{{item_three|[[Video]], [[Electronic Arts]], [[New Media]]}}} |label5 = Birth Place |data5 = {{{item_Four|Najaf, Iraq}}} |label6 = Autor |data6 = {{{item_Five|Slowking4}}} }} <div style="clear:both;"></div> {{Infobox | name = Digital Media Concepts | bodystyle = | titlestyle = | abovestyle = background:#cfc; | subheaderstyle = | title = Test Infobox | above = Above text | subheader = Subheader above image | subheader2 = Second subheader | imagestyle = | captionstyle = | image = [[File:Example-serious.jpg|200px|alt=Example alt text]] | caption = Caption displayed below Example-serious.jpg | headerstyle = background:#ccf; | labelstyle = background:#ddf; | datastyle = | header1 = Header defined alone | label1 = | data1 = | header2 = | label2 = Label defined alone does not display (needs data, or is suppressed) | data2 = | header3 = | label3 = | data3 = Data defined alone | header4 = All three defined (header, label, data, all with same number) | label4 = does not display (same number as a header) | data4 = does not display (same number as a header) | header5 = | label5 = Label and data defined (label) | data5 = Label and data defined (data) | belowstyle = background:#ddf; | below = Below text }} <div style="clear:both;"></div> ==Navigational Templates== [[:w:Wikipedia:Navigation template|Navigation Template on Wikipedia]] == Media Templates == {{Listen | filename = Lion raring-sound1TamilNadu178.ogg | title = Lion roaring | plain = yes | style = float:left }} <div style="clear:both;"></div> == Useful links == * [[Help:Wikitext quick reference|Quick Reference]] * [[:w:Wikipedia:Tutorial/Wikipedia_links|Wikipedia Tutorial]] * [http://www.wikihow.com/Cite How to cite on Wikipedia] * [[:w:Help:Link|Using external links]] == Completed Assignments == {{Subpages/List}} ==References== {{reflist}} [[Category:Art]] [[Category:Culture]] [[Category:Human]] [[Category:Assignments]] [[Category:Digital art]] [[Category:Courses]] [[Category:Education]] qhicjjae9kanxscoibeipole2u0h09f NCERT/Textbook Solutions/Class VII/Geography 0 214229 2803354 2742266 2026-04-07T16:21:35Z ~2026-21400-51 3064352 2803354 wikitext text/x-wiki These syllabus are periodically reviewed and revised. The NCERT book for a particular subject is divided into various chapters and every chapter has a set of questions following the chapter. This section provides answers to the questions at the end of each chapter in the '''Geography''' book, '''Our Environment''', for [[NCERT/Textbook_Solutions#Solutions_to_Textbooks_of_Class_VII |'''Class-VII''']]. ==Chapter 01 == The Questions with Answers of this chapter are provided below:- '''Question 01:''' Answer the following questions:- '''(i)''' What is an ecosystem? '''Answer:Ecosystem is a community of living organisms in conjunction with the nonliving components of their environment (things like air, water and mineral soil), interacting as a system.''' '''(ii)''' What do you mean by natural environment? '''Answer: The environment which is created by nature comprises of land, water, air, plants and animals. This is known as natural environment ''' '''(iii)''' Which are the major components of the environment? '''Answer:''' Two major components of the environment are- 1.natural components-land,water,plant,etc 2.Artifical/man made components - road, building, machine,etc. '''(iv)''' Give four examples of human made environment. '''Answer:''' Few examples of human made environment are: # Parks # Buildings # Roads # Vehicles # Bridges # Industries #monuments '''(v)''' What is lithosphere? '''Answer:''' [[Wikipedia:Lithosphere| Lithosphere]] is the solid crust or the hard top layer of the earth. It includes the crust and the uppermost mantle, which constitute the hard and rigid outer layer of the Earth. '''(vi)''' Which are the two major components of biotic environment? '''Answer:''' Two major components of biotic environment are Plants and Animals. '''(vii)''' What is biosphere? '''Answer:''' Biosphere is a narrow zone of the earth where land, water and air interact with each other to support life. It consists of plant and animal kingdom together. It is a global sum of all ecosystems. '''Question 2.''' Choose the correct answer. '''(i)''' Which is not a natural ecosystem? (a) Desert (b)Aquariumum (c) Forest '''Answer:''' (b) Aquarium '''(ii)''' Which is not a component of human environment? (a) Land (b) Religion (c) Community '''Answer:''' (a) Land '''(iii)''' Which is a human made environment? (a) Mountain (b) Sea (c) Road '''Answer:''' (c) Road '''(iv)''' Which is a threat to environment? (a) Growing plant (b) Growing population (c) Growing crops '''Answer:''' (b) Growing population '''Question 3.''' Match the following. {| style="width:100%;" class="wikitable" border="0" cellpadding="2" cellspacing="2" ! scope="col" width=20%| Col-1 ! scope="col" width=20%| Col-2 |- ! | Biosphere | blanket of air which surrounds the earth |- ! scope="row" | Atmosphere | domain of water |- ! scope="row" | Hydrosphere | gravitational force of the earth |- ! scope="row" | Environment | our surroundings |- ! scope="row" | ----- | narrow zone where land water and air interact |- ! scope="row" | ----- | relation between the organisms and their surroundings |- |} '''Answer:''' {| style="width:100%;" class="wikitable" border="0" cellpadding="2" cellspacing="2" ! scope="col" width=20%| Col-1 ! scope="col" width=20%| Col-2 |- ! | Biosphere | narrow zone where land water and air interact |- ! scope="row" | Atmosphere | blanket of air which surrounds the earth |- ! scope="row" | Hydrosphere | domain of water |- ! scope="row" | Environment | our surroundings |- |} '''Question 4.''' Give reasons. (i) Man modifies his environment (ii) Plants and animals depend on each other '''Answer:''' (i) Man modifies his environment because of his growing needs. He is capable of modifying it according to his need to live a comfortable life. Humans learn new ways to use and change environment and as a result invented many things. Industrial revolution enabled large scale production of goods. Transportation became faster and more comfortable. Information revolution made communication easier and faster across the world. (ii) Plants and animals depend on each other for their sustainability. Animals consume plants for their living and also takes oxygen from them. Few carnivorous animals inturn eat other animals. Plants are dependent on animals as they give out carbon dioxide which is important for photosynthesis. Also, dead remains of animals provide nutrients to the plants. ==Chapter 02 == The Questions with Answers of this chapter are provided below:- '''Question 1.''' Answer the following questions. '''(i)''' What are the three layers of the earth? '''Answer''' The three layers of the Earth are the crust, the mantle and the core. '''(ii)''' What is a rock? '''Answer''' Any natural mass of solid mineral matter that makes up the Earth’s crust is called a rock. '''(iii)''' Name three types of rocks. '''Answer''' The three types of rocks are igneous rocks, sedimentary rocks and metamorphic rocks. '''(iv)''' How are extrusive and intrusive rocks formed? '''Answer''' Extrusive rocks are formed by the molten lava which comes on the earth’s surface and rapidly cools down to becomes solid. When the molten magma cools down deep inside the earth’s crust then the solid rocks so formed are called intrusive rocks. and also is most power full rock '''(v)''' What do you mean by a rock cycle? '''Answer''' When one type of rock changes to another type under certain conditions in a cyclic manner then this process of transformation of the rock from one to another is known as the rock cycle. '''(vi)''' What are the uses of rocks? '''Answer''' The uses of the rocks are as follows : * Hard rocks are used in construction of buildings and roads. * Some rocks are shiny and precious therefore used for making jewellery. * Rocks are made up of different minerals and are very important to humankind. * Some are used as fuels. For example, coal, natural gas and petroleum. * Soft rocks are used for making talcum powder, chalks etc. '''(vii)''' What are metamorphic rocks? '''Answer''' The rocks which are formed due to conversion of igneous and sedimentary rocks under great heat and pressure is called metamorphic rocks. '''Question 2.''' Tick the correct answer. '''(i)''' The rock which is made up of molten magma is (a) Igneous (b) Sedimentary (c) Metamorphic '''Answer''' (a) Igneous '''(ii)''' The innermost layer of the earth is (a) Crust (b) Core (c) Mantle '''Answer''' (b) Core '''(iii)''' Gold, petroleum and coal are examples of (a) Rocks (b) Minerals (c) Fossils '''Answer''' (b) Minerals '''(iv)''' Rocks which contain fossils are (a) Sedimentary rocks (b) Metamorphic rocks (c) Igneous rocks '''Answer''' (a) Sedimentary rocks '''(v)''' The thinnest layer of the earth is (a) Crust (b) Mantle (c) Core '''Answer''' (a) Crust '''Question 3.''' Match the following. {| style="width:100%;" class="wikitable" border="0" cellpadding="2" cellspacing="2" ! scope="col" width=20%| Col-1 ! scope="col" width=20%| Col-2 |- ! | Core | Earth’s surface |- ! scope="row" | Minerals | Used for roads and buildings |- ! scope="row" | Rocks | Made of silicon and alumina |- ! scope="row" | Clay | Has definite chemical composition |- ! scope="row" | Sial | Innermost layer |- ! scope="row" | ----- | Changes into slate |- ! scope="row" | ----- | Process of transformation of the rock |- |} '''Answer''' (i) Cores (e) Innermost layers (ii) Minerals (d) Has definite chemical composition (iii) Rocks (b) Used for roads and buildings (iv) Clay (f) Changes into slate (v) Sial (c) Made of silicon and alumina '''Question 4.''' Give reasons. '''(i)''' We cannot go to the centre of the earth. '''Answer''' We cannot go to the centre of the earth because the it has very high temperature and pressure and lies 6000 km below the ocean floor. We will not able to survive there because there is no oxygen or<graph>{ "version": 2, "width": 400, "height": 200, "data": [ { "name": "table", "values": [ { "x": 0, "y": 1 }, { "x": 1, "y": 3 }, { "x": 2, "y": 2 }, { "x": 3, "y": 4 } ] } ], "scales": [ { "name": "x", "type": "ordinal", "range": "width", "zero": false, "domain": { "data": "table", "field": "x" } }, { "name": "y", "type": "linear", "range": "height", "nice": true, "domain": { "data": "table", "field": "y" } } ], "axes": [ { "type": "x", "scale": "x" }, { "type": "y", "scale": "y" } ], "marks": [ { "type": "rect", "from": { "data": "table" }, "properties": { "enter": { "x": { "scale": "x", "field": "x" }, "y": { "scale": "y", "field": "y" }, "y2": { "scale": "y", "value": 0 }, "fill": { "value": "steelblue" }, "width": { "scale": "x", "band": "true", "offset": -1 } } } } ] }</graph>conditions. favourable '''(ii)''' Sedimentary rocks are formed from sediments. '''Answer''' Sedimentary rocks are formed from sediments because of extreme compression and hardening of the particles of sediment which are transported and deposited by wind, water etc. '''(iii)''' Limestone is changed into marble. '''Answer''' Limestone is changed into marble because of extreme heat and pressure as it is a sedimentary rock. ==Chapter 03 == ==Chapter 04 == ==Chapter 05 Water== ==Chapter 06 Natural Vegetation and Wildlife== ==Chapter 7 == Geography ==Chapter 08 == ==Chapter 09 Life in the Temperate Grasslands== ==Chapter 10 == ==See Also== [[NCERT/Textbook Solutions]] [[NCERT/Textbook Solutions/Class VII/History]] [[NCERT/Textbook Solutions/Class VII/Civics]] ==External Links== * [http://www.ncert.nic.in/index.html Official website of NCERT] * [http://epathshala.nic.in/e-pathshala-4/flipbook/ NCERT textbooks available online] ==Geography ncert== {{Reflist}} [[Category:Geography]] 5sal2m8ej68mmelkgrmcbiy3pwp0re6 Motivation and emotion/Lectures/Individual emotions/Images 0 276193 2803488 2462624 2026-04-08T06:46:23Z Jtneill 10242 /* Images */ + 2803488 wikitext text/x-wiki ==Images== <gallery> File:Emotion Categories.png File:Font Awesome 5 solid car-crash.svg <!-- File:Personality Question 7741.svg --> File:Emotion collage.png File:Scared Girl.jpg File:Angry woman.jpg File:Disgust expression cropped.jpg File:Contempt.jpg <!-- File:PSM V36 D704 Facial expression of contempt.jpg --> File:Sad girl cropped.jpg File:Interest.jpg File:Happiness cropped.jpg File:Taunting 0001 cropped.jpg File:Daddy, what did You do in the Great War cropped.jpg File:Embarrassed woman.jpg File:US Navy 061224-N-9909C-009 A proud father and Sailor from the destroyer USS Halsey (DDG 97) holds his child for the first time.jpg File:Subtle envy.jpg File:Wikimania Volunteers Gratitude Meetup P1050338 cropped.jpg File:1868 Renoir Summer anagoria cropped.jpg File:Regret cropped.jpg File:Hopeful child.jpg File:Schadenfreude.png <!-- File:Smug face cropped.jpg --> File:Evstafiev-bosnia-sarajevo-funeral-reaction cropped.jpg File:US specialist helping Afghan nomads cropped.jpg </gallery> [[Motivation and emotion/Lectures/Individual emotions]] [[Category:Motivation and emotion/Lectures]] 2clzzi3qqca6e1yi981kx04k0yz9gk7 C language in plain view 0 285380 2803332 2803240 2026-04-07T13:53:15Z Young1lim 21186 /* Applications */ 2803332 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260407.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> 7ulw95zcemy8xqyccrg5hqezpj03iuk User:Dc.samizdat/Real Euclidean four-dimensional space R⁴ 2 289273 2803360 2803312 2026-04-07T17:03:01Z Dc.samizdat 2856930 /* Real Euclidean four-dimensional space R⁴ */removed from abstract 2803360 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == Light propagates through 4-space at twice its apparent velocity ''c''== <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (teflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} i42tipgf68yjijecirrn4a2if2oa8e4 2803361 2803360 2026-04-07T17:05:58Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2803361 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == Light propagates through 4-space at twice its apparent velocity ''c''== <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (teflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} kav2mt2s5xii9bd0hcvogy5gxydw1db 2803362 2803361 2026-04-07T17:15:23Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2803362 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == Light propagates through 4-space at twice its apparent velocity ''c''== <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (teflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} p31lda8wcz9yraiy4pjbzh8rvlwfdid 2803363 2803362 2026-04-07T17:17:31Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2803363 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving differently. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == Light propagates through 4-space at twice its apparent velocity ''c''== <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (teflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} qq5ohn9jcrqmdu1gt2swp8gq795y095 2803364 2803363 2026-04-07T17:20:42Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2803364 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == Light propagates through 4-space at twice its apparent velocity ''c''== <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (teflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} cbb3gpqzh4defbouv2rnn0jviqsfbro 2803365 2803364 2026-04-07T17:43:17Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803365 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == Light propagates through 4-space at twice its apparent velocity ''c''== <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (reflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 37mzcvdilq5l0c5dgwkk37lqg1i97r0 2803368 2803365 2026-04-07T18:17:15Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete reflections and rotations */ 2803368 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete reflections and rotations == <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (reflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} fwqvpedvlj536211rukudsb9zwqicf2 2803382 2803368 2026-04-07T18:57:18Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete reflections and rotations */ 2803382 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete reflections and rotations == Coxeter theory describes all the possible motions of an object in space as simple commutative functions of its discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a finite number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be described as the object propagating itself in space by a discrete set of self-reflections. <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> ...to readers who have not studied Coxeter (almost all readers including TAC), this section is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood, the "math" here is really just simple pictures (reflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} gaxeqfoygzjjsm31gd69su2h9vpwgs1 2803406 2803382 2026-04-07T20:38:15Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete reflections and rotations */ 2803406 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) could be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say clearly what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} nzzl4sd787010q4wmpe67hj9nec7511 2803408 2803406 2026-04-07T20:39:58Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2803408 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations here geometrically: I could say clearly what they mean in spatial terms in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations)...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 5sq5ehld1sciv09svl6otcjsla1nyk8 2803414 2803408 2026-04-07T20:52:40Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2803414 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do that here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} q9qcwxifkkff6l589o8yxsnqa7f51cp 2803415 2803414 2026-04-07T20:53:43Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2803415 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two completely orthogonal double reflections in non-intersecting pairs of parallel planes at once, a reflection in four non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} emhgzng2safp0ouowb1fd67bc0ctwtd 2803427 2803415 2026-04-07T21:12:27Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803427 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ijx94xtqnqwbrhycg4ni6ohrc264z3q 2803428 2803427 2026-04-07T21:13:31Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803428 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 6yxb9xt1okq6oqau2qblpmk55h2kvf1 2803430 2803428 2026-04-07T21:20:16Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803430 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved. The total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ie2ff8qp4e6neb7wf9dgbkygpkg43et 2803431 2803430 2026-04-07T21:22:28Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803431 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of completely orthogonal reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} l7hh94m2bu8ot0bmi0ab3fw35emfi4a 2803432 2803431 2026-04-07T21:24:18Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803432 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} dgbmazy1bj72y7qxwhfyuqwhj3n4l36 2803433 2803432 2026-04-07T21:26:55Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803433 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two completely orthogonal translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 7cy4c3ubiytsm7yq9rkpzcxbvltziqp 2803434 2803433 2026-04-07T21:31:34Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803434 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} lc2ydxcg3u8h3htx6ven6gw1y8lkkpc 2803435 2803434 2026-04-07T21:39:29Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2803435 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before i can publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 887lpr9r1n388robsdvhdhce9zj7ljs 2803436 2803435 2026-04-07T21:41:33Z Dc.samizdat 2856930 /* Distribution of stars in our galaxy */ 2803436 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} slj0ozadwdodgeppagh7qyxih2qrx43 2803437 2803436 2026-04-07T21:43:33Z Dc.samizdat 2856930 /* Rotations */ 2803437 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of the discrete isoclinic rotation characteristic of the particle and some regular 4-polytope. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} fujyqyrjgu8wq04xfalrigwgf32mcn0 2803439 2803437 2026-04-07T21:44:57Z Dc.samizdat 2856930 /* Rotations */ 2803439 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia, within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} r1qyibwahr0qju4vomnalp0yx5ijjql 2803440 2803439 2026-04-07T21:52:09Z Dc.samizdat 2856930 /* Rotations */ 2803440 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as a set of simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} l80zy9hj6mvqofryxdpfjaiki62fc5b 2803441 2803440 2026-04-07T22:49:50Z Dc.samizdat 2856930 2803441 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description should be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} e0ays71aqelfv39mswf1mojjz4dph7t 2803442 2803441 2026-04-07T22:50:27Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2803442 wikitext text/x-wiki = Real Euclidean four-dimensional space R⁴ = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - March 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. == A theory of the Euclidean cosmos == The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move through Euclidean 4-space at velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical spiral galaxy such as ours is a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres. ...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy... The galaxy as a whole, or more properly its orbital center point, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the cylinder; their trajectories are screw-displacements. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis. It is the tremendous inertial force of stars in motion at velocity <math>c</math> that holds the cylinder of motion together. The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. For all observers, the conjectured origin point of the universe corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space (the same point in the same Euclidean 4-space for all observers). The big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured separately, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold that is such an evolving surface boundary is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity <math>c</math> of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know nothing about its interior 4-ball. Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity <math>c</math>. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from atoms to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two orthogonal motions each at the constant universal rate of transformation <math>c</math>. Every object moves on its own distinct geodesic spiral. Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity <math>c</math> in all four dimensions concurrently, so they move diagonally through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c^\prime = 2c</math>, so we can see in 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math> in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <small><math>\mathrm{Q}</math></small> denote a rotation, <small><math>\mathrm{R}</math></small> a reflection, <small><math>\mathrm{T}</math></small> a translation, and let <small><math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math></small> denote a product of several such transformations, all commutative with one another. Then <small><math>\mathrm{RT}</math></small> is a glide-reflection (in two or three dimensions), <small><math>\mathrm{QR}</math></small> is a rotary-reflection, <small><math>\mathrm{QT}</math></small> is a screw-displacement, and <small><math>\mathrm{Q^2}</math></small> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r</math></small><br> where <small><math>(2^q + r \le n)</math></small>, the number of dimensions.<br> Transformations involving a translation are expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math></small><br> where <small><math>(2^q + r + 1 \le n)</math></small>.<br> For <small><math>(n = 4)</math></small> in particular, every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <small><math>\mathrm{Q^2}</math></small> or a <small><math>\mathrm{QT}</math></small>, because we can view any <small><math>\mathrm{QT}</math></small> as a <small><math>\mathrm{Q^2}</math></small> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <small><math>\mathrm{Q^2}</math></small>. By the same principle, we can view any <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> as an isoclinic (equi-angled) <small><math>\mathrm{Q^2}</math></small> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<small><math>\mathrm{T}</math></small>) for ''one'' of the two rotations (<small><math>\mathrm{Q}</math></small>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<small><math>\mathrm{Q}</math></small>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<small><math>\mathrm{T}</math></small>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <small><math>SO(4)</math></small> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <small><math>SO(4)</math></small> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, we shall see that stationary atoms exhibit the <small><math>SO(4)</math></small> symmetries of the discrete isoclinic (equi-angled) double rotations (<small><math>\mathrm{Q^2}</math></small>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity ''c'', in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity ''c'' through the universe, which is real Euclidean 4-dimensional space <small><math>\mathbb{R^4}</math></small>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity ''c''. Their direction of movement through space may be different for different observers (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, may be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to ''c'', it is the same prediction in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" ''c'', although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space. ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords. Euclidean relativity is not even a fringe theory; no physicists have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c^\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number <small><math>\sqrt{4}</math></small>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, leaving itself right-side-out again. Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<small><math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math></small><br> where <small><math>(2^q + r + t \le 4)</math></small>. Every displacement is either a double rotation <small><math>\mathrm{Q}^2</math></small>, or a screw-displacement <small><math>\mathrm{QT}</math></small> [where the rotation component <small><math>\mathrm{Q}</math></small> is a simple rotation, but the <small><math>\mathrm{QT}</math></small> is chiral like a <small><math>\mathrm{Q^2}</math></small>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <small><math>\mathrm{QRT}</math></small>.</blockquote> While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. <small><math>\mathrm{R}, \mathrm{T}, \mathrm{Q}</math></small> are just what they are in three-dimensional space, but <small><math>\mathrm{Q}^2</math></small> is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in. ...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is "just math", not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the "math" here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here... == Light propagates through 4-space at twice its apparent velocity ''c''== Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <small><math>\mathrm{R}^4</math></small>, which may be termed a double translation <small><math>\mathrm{T}^2</math></small>, a pure translation via two pairs of parallel reflections, without any rotation component <small><math>\mathrm{Q}</math></small>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <small><math>\mathrm{QT}</math></small>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <small><math>\mathrm{Q^2}</math></small>, an isoclinically rotating object such as an atom. A simple rotation <small><math>\mathrm{Q}</math></small> or simple translation <small><math>\mathrm{T}</math></small> is a double reflection <small><math>\mathrm{R^2}</math></small>, so a <small><math>\mathrm{QT}</math></small> or <small><math>\mathrm{Q^2}</math></small> is also an <small><math>\mathrm{R^4}</math></small>, but not with the same group of reflection angles as a light signal <small><math>\mathrm{R^4}</math></small>. A translation <small><math>\mathrm{T = R^2}</math></small> is a double reflection in two parallel planes, and a rotation <small><math>\mathrm{Q = R^2}</math></small> is a double reflection in two intersecting planes, as in a <small><math>\mathrm{QT = R^4}</math></small> which is both at once. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a <small><math>\mathrm{T^2}</math></small> all the motion goes to translation, so the translation goes twice as far as the simple translation <small><math>\mathrm{T}</math></small> in a <small><math>\mathrm{QT}</math></small>. A double translation <small><math>\mathrm{T^2 = R^4}</math></small> is the opposite of a double rotation <small><math>\mathrm{Q^2 = R^4}</math></small>, which is stationary but rotates twice as fast as the simple rotation <small><math>\mathrm{Q}</math></small> in a <small><math>\mathrm{QT}</math></small>. The product of the two translations in a <small><math>\mathrm{T^2}</math></small> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <small><math>\mathrm{T}</math></small> over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 4-hypercube (also known as the 8-cell or tesseract) is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon. So its long diameter (twice its radius) is exactly twice its edge length.}} The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <small><math>\mathrm{T^2}</math></small> cannot reposition a 4-polytope the way a <small><math>\mathrm{QT}</math></small> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter. ...lensing of double translations <small><math>\mathrm{T^2 = R^4}</math></small> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet... == The Kepler problem is framed in Euclidean 4-space == The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the <small><math>SO(4)</math></small> symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935<ref>V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.</ref> and Moser in 1970<ref>J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636</ref> observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. ... Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q<sup>2</sup>) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity <math>c</math>, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity <math>c</math>. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q<sup>2</sup>) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q<sup>2</sup>) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T). All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be. The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space... ...cite Jesper Goransson's very concise paper ... Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of <small><math>SO(3)</math></small> and <small><math>\mathbb{R^3}</math></small>, in contrast to the <small><math>SO(4)</math></small> symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space <small><math>\mathbb{R^3}</math></small>, spherical space <small><math>S^3</math></small> and Euclidean space <small><math>\mathbb{R^4}</math></small>. Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an <small><math>SO(4)</math></small> rotation symmetry corresponding to an isoclinic double rotation (<small><math>\mathrm{Q^2}</math></small>). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity <math>c</math> and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (<small><math>\mathrm{QT}</math></small>) at constant velocity <math>c</math>. From this viewpoint the atom has only a simple <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>), breaking its stationary <small><math>SO(4)</math></small> isoclinic rotation symmetry (<small><math>\mathrm{Q^2}</math></small>). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity <math>c</math> is very narrow: only the diameter of the rotating atom itself. The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same <small><math>SO(4)</math></small> rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (<small><math>\mathrm{T}</math></small>) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity <math>c</math>. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant. ...Its Kepler orbit around the sun is its <small><math>SO(3)</math></small> rotation component (<small><math>\mathrm{Q}</math></small>). Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (<small><math>\mathrm{QT}</math></small>) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (<small><math>\mathrm{T}</math></small>) is a constant of motion of the entire system in both cases. ...Anco and Maghadam found that <small><math>SO(4)</math></small>) breaks to ... <small><math>S^3</math></small>)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small>) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). ... Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom... ... As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space <small><math>SO(4)</math></small>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. ...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper... == Rotations == The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time. For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]]. In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, "frustrated" by their twisted geometry. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle. == A theory of the Euclidean atom == ... == Light and Mass are Reflection and Rotation == The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively. ... Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry. Light is .... Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]]. ... === Atoms are 4-polytopes === ... == Relativity in real space of four or more orthogonal dimensions == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, and any orthogonal 4-manifold. Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics. Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light. === Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions === ...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here... Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery. === General relativity is Galilean relativity in a general space of four orthogonal dimensions === ... == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the degree of dimensional analogy of which they are capable, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity ''c'', with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <small><math>SO(4)</math></small> breaks to ... <small><math>S^3</math></small>... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... <small><math>H^3</math></small> ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space. ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others. The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>){{spaces|3}}(–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,–<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90<sup>o</sup> bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60<sup>o</sup> bend, or as a straight line with one 60<sup>o</sup> bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V<sub>0</sub> in the original great hexagon plane of isoclinic rotation P<sub>0</sub>, the first vertex reached V<sub>1</sub> is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P<sub>1</sub>. P<sub>1</sub> is inclined to P<sub>0</sub> at a 60° angle.{{Efn|P<sub>0</sub> and P<sub>1</sub> lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V<sub>2</sub> is 120 degrees beyond V<sub>1</sub> along a second {{radic|3}} chord lying in another hexagonal plane P<sub>2</sub> that is Clifford parallel to P<sub>0</sub>.{{Efn|P<sub>0</sub> and P<sub>2</sub> are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V<sub>0</sub> and V<sub>2</sub> are ''two'' {{radic|3}} chords apart{{Efn|V<sub>0</sub> and V<sub>2</sub> are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V<sub>0</sub> and V<sub>2</sub> are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P<sub>0</sub> and P<sub>2</sub> are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V<sub>1</sub> lies in both intersecting planes P<sub>1</sub> and P<sub>2</sub>, as V<sub>0</sub> lies in both P<sub>0</sub> and P<sub>1</sub>. But P<sub>0</sub> and P<sub>2</sub> have ''no'' vertices in common; they do not intersect.) The third vertex reached V<sub>3</sub> is 120 degrees beyond V<sub>2</sub> along a third {{radic|3}} chord lying in another hexagonal plane P<sub>3</sub> that is Clifford parallel to P<sub>1</sub>. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V<sub>0</sub> to V<sub>3</sub> is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹<sub>4</sub>.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} qavlht0yhavqexi4c9patdmf0gs7bwe Global Audiology 0 292077 2803371 2746172 2026-04-07T18:19:48Z RadiX 155307 +button 2803371 wikitext text/x-wiki {{:Global Audiology/Header}} <!--- == DRAFT == To be developed once license issues are addressed https://globalaudiology.org/about/index ---> According to the [https://www.who.int/teams/noncommunicable-diseases/sensory-functions-disability-and-rehabilitation/highlighting-priorities-for-ear-and-hearing-care World Report on Hearing] produced by the [[:en:w:World Health Organization|World Health Organization]] (WHO), more than 1.5 billion people will experience some decline in their hearing capacity during their lifetime, of whom at least 430 million will require hearing care. Moreover, it is suggested that over 80% of this population lives in low and middle-income countries where there is very little access to any audiological services. Hence, there is much work to be done in audiology to bridge this inequality in education and access to services. '''Global Audiology''' is a portal that is aimed to provide an understanding of audiology education and practice around the world. The goal of the information provided in this portal is to facilitate networking among stakeholders in hearing health as well as help promote the development of audiology in settings where fewer resources are available. Developing knowledge about current practice trends is the first step in helping to standardize audiology practice and ultimately improve audiological care, facilitate the standardization of audiology practice, and help with providing better access to audiology services. Drs. [https://www.vinayamanchaiah.com/ Vinaya Manchaiah] and [https://profiles.utdallas.edu/roeser Ross Rosser] co-developed this project in 2016 and created the website and content with help from volunteers across the globe. However, the Global Audiology initiative was endorsed and since 2019 it is managed by the [[:en:w:International Society of Audiology|International Society of Audiology]] (ISA). The Global Audiology Working Group within the ISA oversees the project. Initially, the content was on its own website, and in 2023 the content was moved to Wikiversity with help from Drs. [https://www.linkedin.com/in/thaiscmorata10 Thais Morata], [https://br.linkedin.com/in/alexandre-montilha-2b75b619b/en?trk=people-guest_people_search-card Alexandre Montilha], and [http://linkedin.com/in/joyce-rodvie-sagun-4691bb182 Joyce Rodvie Sagun]. The move was motivated by a desire to provide mechanisms to facilitate input from our communities of interest. {{HTitle|Call to Action}}All audiologists and hearing care professionals are invited to join Global Audiology! This project is a collaborative effort to create a comprehensive and up-to-date resource for audiology information worldwide. We need your help to make this project a success. The Global Audiology initiative relies on committed volunteers for the development and management of the content. The Global Audiology working group within the ISA, with input from the ISA [https://isa-audiology.org/about-isa/the-executive Executive Committee] members, developed the project structure and processes. The Global Audiology Working Group moderates the content within the Wikiversity pages. However, the information for the country-specific page is created by volunteers. A few people will take the lead in developing much of the content to ensure consistency in writing. However, anyone can edit to develop the information further and ensure the content is accurate and up-to-date. Start contributing to Global Audiology by clicking on the [https://en.wikiversity.org/wiki/Global_Audiology/Help How to Edit] tab. === How else can you contribute? === *Write an article about audiology in your home country. If you do not see a country-specific page to which you would like to contribute, visit the [https://en.wikiversity.org/wiki/Global_Audiology/Help How to Edit] section for a step-by-step guide or contact [https://isa-audiology.org/about-isa/working-groups ISA administration] to have the page created. *Share your expertise by answering other audiologists, audiology students, and hearing healthcare professionals' questions. *Promote Global Audiology to your colleagues and friends. *Share audiology-related events and other initiatives. === Your contribution will make a difference === Global Audiology is a valuable resource for audiologists around the world. By contributing to this project, you can help improve the quality of care for people with hearing and vestibular disorders. {{:Global Audiology/Input}} {{HTitle|Useful Audiology Resources and Links}}* [https://meta.wikimedia.org/wiki/WikiProject_Hearing_Health| WikiProject Hearing Health] * [https://player.vimeo.com/video/478861791?h=e4cbc7d534| Global audiology open access webinar, International Society of Audiology] * [https://www.asha.org/ American Speech-Language-Hearing Association (ASHA)] * [https://www.audiologyonline.com/ Audiology Online] * [https://www.who.int/campaigns/world-hearing-day World Health Organization (WHO) World Hearing Day] * [https://www.nidcd.nih.gov/health/hearing-ear-infections-deafness National Institute on Deafness and Other Communications Disorders (NIDCD)] * [https://www.cdc.gov/hearingloss/default.html Centers for Disease Control and Prevention (CDC Hearing Loss)] * [https://www.cdc.gov/niosh/topics/noise/default.html The National Institute for Occupational Safety and Health (NIOSH)] * [https://www.osha.gov/noise Occupational Safety and Health Administration (OSHA)] * [http://www.jcih.org/ Joint Committee on Infant Hearing (JCIH)] * [https://www.audiology.org/ The American Academy of Audiology (AAA)] * [https://www.infanthearing.org/about/ National Center for Hearing Assessment and Management (NCHAM)] * [https://computationalaudiology.com/introduction-to-computational-audiology/ Computational Audiology Network (CAN)] * [https://www.thebsa.org.uk/ British Society of Audiology (BSA)] * [https://portal.acaud.com.au/ Australian College of Audiology] * [https://audiology.org.nz/ New Zealand Audiological Society] * [https://coalitionforglobalhearinghealth.org/ Coalition for Global Hearing Health] </div></div> [[Category:Healthcare]] [[Category:Audiology]] [[Category:Otorhinolaryngology]] [[Category:Communication]] 1b17sxdnoav8a8eyvdia7dalockfe26 Global Audiology/Help 0 292742 2803370 2803174 2026-04-07T18:19:08Z RadiX 155307 2803370 wikitext text/x-wiki {{:Global Audiology/Header}} You can contribute to the online information about audiology services and practices worldwide by creating content on Global Audiology at Wikiversity. Share your knowledge and experiences to help others learn more about audiology. Your contributions to Wikiversity will help ensure that everyone has access to reliable information about audiology services and practices. Here is a guide for adding or editing content on audiology practices on Wikiversity. Whether you're a Wikimedia user or not, we have prepared steps for you. We also provide resources and tips to help you create and edit content (located in the Resources section). Tutorials specific to editing Wikipedia are at the bottom of this page. We hope this guide can help you get started on your contribution. We welcome all contributions, no matter how small. We are always open to feedback, so please let us know if you have any questions or suggestions. Thank you for your interest in contributing to Global Audiology at Wikiversity! {{HRow}} ===Suggested article structure=== The following is the suggested article structure for a country-specific page. Working on a small group preferably working closely with the local audiology society or a professional body (e.g., [https://isa-audiology.org/affiliates/overview ISA's Affiliated Societies]) is suggested to ensure the country specific page will include comprehensive information. Also, providing links to relevant websites as well as adding media (images, videos) is likely to enhance reading experience of the content. *Brief Country Information *History of Audiology and Aural Care *Incidence and Prevalence of Hearing Loss *Hearing Care Services **Professionals providing hearing care services **Audiological services **Services offered by Otolaryngologists, Otologists, and Otoneurologists **Role of primary health care providers and community health workers in hearing care **Laws related to hearing care services *Education and Professional Practice **Education of professionals working in hearing care services **Professional and Regulatory Bodies **Scope of Practice and Licensing *Audiology Research *Audiology Charities *Challenges and Opportunities *Acknowledgments *References *Author Information {{HRow}} ===Step-by-step=== ====For Wikimedia Users==== ===== Step 1: Create an account ===== Before creating content on Wikiversity, you need to create a Wikimedia account. If you already have an account, you can skip this step and move on to step 2. Go to the Wikimedia homepage click "Create account" in the upper right-hand corner, and follow the prompts to create your account (or directly to [https://meta.wikimedia.org/w/index.php?title=Special:CreateAccount&returnto=Main+Page Create Account]. You must provide your username and email address and choose a password. ===== Step 2: Familiarize yourself with the community guidelines ===== Before contributing to Global Audiology at Wikiversity, take the time to review existing entries and read the community guidelines. This will help you understand what content is acceptable and what isn't and help you avoid mistakes that could result in your contributions being edited or deleted. For instance, the language is neutral, the content you develop is politically neutral, does not celebrate one individual, company, or institution, and instead presents a more balanced view of the structure and status of audiology in the country or region. ===== Step 3: Visit the Global Audiology Homepage ===== The Global Audiology Homepage provides a wealth of information about the project's mission and purpose, as well as resources and other materials related to hearing health and audiology. Check out the [https://en.wikiversity.org/wiki/Global_Audiology homepage] and explore to learn more about our initiatives and how you can get involved. ===== Step 4: Conduct research ===== Find the Global Audiology content you want to add to or modify. Prepare your content by doing some research on your chosen topic. Use reliable sources, such as peer-reviewed academic journals or bona fide websites and/or news services. We suggest you include references to key statements. It is easy, Wikimedia formats it for you). However, it may be difficult to find references to all the content. ===== Step 5: Create your content ===== As soon as you have gathered your research, start creating content. Click on the "Edit" button at the page's top, and using the pencil icon, select “Visual editor”. The toolbar lets you format your text and add headings, links, and other formatting elements. You can edit or add new text, images, or multimedia and edit existing content. The article structure should be followed and your content should be neutral, well-organized, easy to read, accurate, relevant, and properly cited (more training resources are in the Resources section). ===== Step 6: Preview and save ===== Before you publish your changes, preview them to ensure they look as you intended. Click the "Apply changes" button and choose the "Show preview" option. You will be able to see how your changes will look before they are actually published. If you need to make further changes to the content, click on the "Edit" button again. To discuss your changes with other contributors or get feedback, click the "Discussion" tab. ===== Step 7: Publish ===== Once you are satisfied with your changes, click "Publish". Please describe succinctly what you have done (created content, added citation, added category, added hyperlink, etc.). Well done! You have successfully added or modified content for the Global Audiology Wikiversity! Continue to update and improve the content over time based on feedback and changes in the field of audiology. Also, consider joining the Discussion page to connect with other Global Audiology contributors. Volunteer subject matter editors and Global Audiology representatives will be alerted of your edit and review it. They might suggest edits or contact you if they have questions. ===== Step 8. More publishing ===== With style modifications, your contribution could become an article for a peer-reviewed journal, such as [https://www.tandfonline.com/doi/full/10.1080/14992020701770843 Audiology in Brazil], [https://www.tandfonline.com/doi/abs/10.3109/00206097409071699 Audiology in Greenland], [https://www.tandfonline.com/doi/abs/10.3109/00381796809075452 Audiology in India], [https://www.tandfonline.com/doi/full/10.1080/14992020500485650 Audiology in South Africa], [https://www.tandfonline.com/doi/full/10.1080/14992020802203322 Audiology education and practice from an international perspective], and others. ====For Non-Wikimedia Users==== ===== Step 1: Visit the Global Audiology homepage ===== The Global Audiology Homepage provides a wealth of information about the project's mission and purpose, as well as resources and other materials related to hearing health and audiology. Check out the [https://en.wikiversity.org/wiki/Global_Audiology homepage] and explore to learn more about our initiatives and how you can get involved. ===== Step 2: Familiarize yourself with the community guidelines ===== Before contributing to Global Audiology at Wikiversity, take the time to read the community guidelines. This will help you understand what content is acceptable and what isn't, and help you avoid mistakes that could result in your contributions being edited or deleted. For instance, the language is neutral, the content you develop is politically neutral, does not celebrate one individual, company, or institution, and instead presents a more balanced view of the status of audiology in the country or region. ===== Step 3: Conduct research ===== Find the Global Audiology content you want to add or modify. Prepare your content by doing some research on your chosen topic. Use reliable sources, such as peer-reviewed academic journals or reputable websites and/or news services. We suggest you include references to key statements and facts. It is easy, Wikimedia formats it for you. However, it may be difficult to find references to all the content. ===== Step 4: Create your content ===== As soon as you have gathered your research, start creating content. Your article can be formatted in Word or PDF. Article structure should be followed, and your content should be neutral, well-organized, easy to read, accurate, relevant, and properly cited. ===== Step 5: Submit your article ===== Please send your article to the Global Audiology team or ISA administrators, so we can publish it on Wikiversity for you. {{:Global Audiology/Input}} === Recently Added Countries === <DynamicPageList> category=New Global Audiology Pages ordermethod=created order=descending count=20 </DynamicPageList> {{HRow}} ===Notes=== * For other Wikiversity training resources, check-out the [https://dashboard.wikiedu.org/training dashboard]. * If the country you wish to contribute to or write an article about is not yet available on Global Audiology at Wikiversity, please reach out to our Global Audiology team or ISA administrators. Let us know what country you would like to be added and we will create it for you. * If you teach audiology, you can ask your students to work on this project, as many others have done for audiology content. Students usually find this activity very motivating. Some examples specific to audiology include: ;Various universities outside the US and Canada: [https://outreachdashboard.wmflabs.org/campaigns/hearing_health__20222024/programs Hearing health campaign 2022-2024] ;University of the Witwatersrand, South Africa: [https://outreachdashboard.wmflabs.org/courses/University_of_the_Witwatersrand/Pathology_of_the_ear_(2nd_Semester)/home Pathology of the ear] ;University of Montreal, Canada: [https://dashboard.wikiedu.org/courses/University_of_Montreal/Promotion_and_prevention_in_audiology-Promotion_et_pr%C3%A9vention_en_audiologie_(Winter-Spring) Promotion and prevention in audiology] ;University of Northern Colorado, USA: [https://dashboard.wikiedu.org/courses/University%20of%20Northern%20Colorado/Hearing%20Loss%20Prevention%20(Fall%20Semester) Hearing loss prevention] {{HRow}} ===Tutorials=== [[File:GLAM logo transparent.png|right|150 px]] This step-by-step [[w:WP:GLAM/TCMI/MAP|guide]] brings together some of the best resources to help you get started in Wikipedia. It is based on a guide originally created by [[w:User:LoriLee|LoriLee]] for middle and high school students to edit Wikipedia. If they can do it, you can! If you would like more general information on why you should contribute to Wikipedia, please see [https://outreachdashboard.wmflabs.org/training training library] ===Training video=== [[File:Editing Wikipedia.webm|Editing Wikipedia]] {{:Global Audiology/footer}} {{HRow}} ===Suggested themes=== Did not find the topic you are looking for? Please let us know: {{Clickable button 2|CONTACT US|url=mailto:contact@globalaudiology.org}} mx84vhmppoj5jzbvmfmbl9k9n7j0gev Motivation and emotion/Book/2023/Flourishing in the elderly 0 296711 2803454 2803279 2026-04-08T02:39:52Z Jtneill 10242 /* Overview */ 2803454 wikitext text/x-wiki {{title|Flourishing in the elderly:<br>How can psychological flourishing be supported in the elderly?}} {{MECR3|1=https://youtu.be/yUAuPhv5S4o}} __TOC__ ==Overview== <br> {{RoundBoxTop|theme=13}} [[File:Optic_stub.svg|thumb|157x196px|'''Figure 1'''. ''Sarah ...''. (image no longer visible after April 2025 due to Wiki Commons feedback and user request for deletion)]] '''''Can you relate''''' ... Sarah, aged 85, begins her day in a house echoing with memories. Faded photographs on the mantelpiece showcase her once pivotal role in the community. From hosting gatherings to being an active voice at the local council, she had always championed causes dear to her. But now, an overwhelming quietness envelops her heart, and she has lost touch with her sense of purpose. Time has seen her closest friends move or pass away. Family visits, once frequent and filled with laughter, have become increasingly rare. This growing isolation weighs on Sarah, exacerbated by a society that seems to prioritise youth over experience. The fast-paced technological world further alienates her; smart devices and social media platforms feel foreign, exacerbating her deep-seated fear of irrelevance in a world that is rapidly evolving without her. In a society increasingly centred on youth, how can seniors like Sarah reclaim their vitality and sense of meaning? {{RoundBoxBottom}} <br> [[File:Optic_stub.svg|thumb|250x250px|'''Figure 2'''. ''An elderly couple ...'' (image no longer visible after March 2026 due to Wiki Commons feedback and user request for deletion)]] The [[wikipedia:Ageing#Successful_ageing|ageing]] population represents an invaluable repository of [[wikipedia:Wisdom|wisdom]], [[wikipedia:Experience|experience]], and [[wikipedia:Insight|insight]] (see Figure 2). Yet, many elderly individuals, much like Sarah, face challenges related to purpose, meaning, and overall psychological [[wikipedia:Well-being|well-being]]. This raises a question; can the elderly attain a state of psychological [[Motivation and emotion/Book/2018/Flourishing|flourishing]], even in the face of age-related adversities? The domain of [[wikipedia:Positive_psychology|positive psychology]] offers evidence-based strategies. Read on to explore what these strategies are and how they can be applied to support seniors in their journey toward a life filled with purpose, meaning, and joy. Specifically, psychological flourishing in the elderly may be supported through a combination of the following four elements, especially when tailored to individual needs and preferences: # [[wikipedia:Social_support|Social engagement]] and meaningful activities such as community involvement or [[wikipedia:Hobby#Psychological_role|hobbies]] (Helliwell et al., 2013) # Mental stimulation via [[wikipedia:Brain_training|cognitive training]] programs can sustain mental acuity and may delay cognitive decline (Ball et al., 2002) # [[Motivation and emotion/Book/2013/Motivating the elderly to exercise|Physical activity]] using exercises tailored to the elderly improve mood and cognitive function (Colcombe & Kramer, 2003) # Incorporating [[Motivation and emotion/Book/2020/Coping and emotion|positive coping strategies]] from positive psychology techniques, such as gratitude exercises, builds [[wikipedia:Psychological_resilience|resilience]] and satisfaction (Seligman et al., 2005) <br> {{RoundBoxTop|theme=3}} [[File:Optics stub.png|right|60px|]] '''Focus questions:''' * What is psychological flourishing? * Why is psychological flourishing important for the elderly? * How does positive psychology foster psychological flourishing? * What can seniors do to flourish? {{RoundBoxBottom}} <br> == What is psychological flourishing? == Psychological flourishing, a term frequently linked to positive psychology, has gained prominence over the years. For elderly individuals, understanding and achieving psychological flourishing is paramount, considering the myriad challenges they confront during this stage of life (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theory, such as self-determination theory and resilience theory, combine with empirical evidence to underpin the suggested four-element model. It is also essential to define and distinguish psychological flourishing from broader notions of general flourishing and well-being. Through understanding this distinction, the unique facets of flourishing and its potential impact on the lives of older individuals become evident. Common misconceptions about psychological flourishing are also addressed, ensuring clarity in subsequent discussions. Psychological flourishing, a concept closely associated with positive psychology, has gained prominence in recent years. For elderly individuals, comprehending and attaining psychological flourishing is crucial, given the diverse challenges encountered in this life stage (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theories such as self-determination theory and resilience theory, as well as empirical evidence, will be used to establish a four-element model of psychological flourishing. Additionally, it is vital to delineate psychological flourishing from broader concepts of general flourishing and well-being. By grasping these distinctions, the unique facets of flourishing and their potential transformative impacts on the lives of older individuals become evident. Common misconceptions about psychological flourishing will also be addressed to ensure clarity in the discussions that follow. === Definitions and distinctions === Psychological flourishing, often simply termed "flourishing", signifies a pinnacle in human functioning. It represents a well-being state that extends beyond merely being free from distress or psychological issues. More than mere survival, flourishing connotes thriving, excelling, and feeling an intense sense of purpose and contentment (Seligman et al., 2005). This condition transcends mere happiness or the absence of mental illness (Sin & Lyubomirsky, 2009). Flourishing provides a holistic view, emphasising positive human functioning across various areas, from relationships to personal growth and purpose. It captures positive emotions, a sense of engagement, strong social connections, and a profound understanding of life's meaning. Distinct from general well-being, flourishing underscores not just feeling good but also functioning effectively, accentuating both [[wikipedia:Hedonic_motivation|hedonic]] (feelings about life) and [[wikipedia:Eudaimonia|eudaimonic]] (functioning in life) well-being facets (Keyes, 2007). Positive psychology’s well-being theory, also known as the PERMA model, conceptualises well-being as a state of comfort, health, or happiness (Seligman, 2011). The model comprises five elements: positive emotion, engagement, positive relationships, meaning, and accomplishment (Seligman, 2011). However, psychological flourishing transcends the ‘feeling good’ aspect of well-being. It encompasses optimal functioning in daily life, incorporating the key principles of self-determination theory: autonomy, relatedness, and competence. === Importance in the context of ageing === Ageing often elicits feelings of apprehension and resignation, influenced by perceptions of decline or limitation, both physically and mentally. Nevertheless, research indicates that many elderly individuals experience periods of growth, insight, and enhanced well-being. For example, the Adult Development and Enrichment Project (ADEPT), conducted by researchers at Pennsylvania State University, underscored the potential for intellectual and emotional growth even in advanced age (Hultsch et al., 1999). Psychological flourishing becomes paramount in this context, assisting the elderly in navigating challenges, capitalising on opportunities, and making meaningful contributions to their communities and personal lives, thereby maintaining their autonomy, relatedness, and competence. Psychological flourishing challenges these notions of decline or limitation, highlighting the profound possibilities for growth, enrichment, and depth in later life (Paterson & Warburton, 2010). Embracing flourishing can significantly transform the way elderly individuals perceive their later years, encouraging them to view this period as an opportunity for renewed purpose, deepened relationships, and the cultivation of new passions or interests. Furthermore, as elderly individuals inevitably face challenges, resilience theory provides a framework for developing a mindset rooted in flourishing (Southwick & Charney, 2012; Bonanno et al., 2004). Such a capacity and mindset act as a solid foundation, empowering seniors to approach obstacles with resilience and poise (Seligman et al., 2005). === Misconceptions === Several misconceptions exist regarding the concept of psychological flourishing. Contrary to common belief, it does not imply a perpetual state of happiness or a life devoid of adversity. Psychological flourishing is not about the absence of negative emotions or challenges. Instead, it focuses on cultivating psychological tools and strategies to navigate adversity, empowering individuals to thrive amidst difficulties (Carstensen et al., 1999). Clarifying these misconceptions paves the way for a deeper understanding of how older individuals can achieve and maintain psychological flourishing. To truly grasp psychological flourishing in older adults, it is crucial to debunk prevalent myths about the ageing process. If left unchallenged, these misconceptions could deter the implementation of strategies promoting well-being in later life (Carstensen et al., 1999; Charles & Carstensen, 2010). Table 1 presents evidence-based counters to these myths, highlighting the potential for growth and vitality among the elderly. {| class="wikitable" |+ Table 1. ''Dispelling Common Misconceptions about the Ageing Process''<br> |- ! Myth !! Fact |- | Ageing leads to inevitable cognitive and emotional decline. || Ageing can offer growth opportunities with the right strategies. |- | Ageing means mental and physical decline. || Many older adults maintain high cognitive and physical activity levels with suitable exercises (Park et al., 2002; Hultsch et al., 1999). |- | Social withdrawal is an ageing inevitability. || While social circles might decrease in size, relationship quality often improves, and staying socially engaged is beneficial (Carstensen et al., 1999; Charles & Carstensen, 2010). |- | Physical activity is risky for older adults. || Moderate physical activity improves mental and physical health in the elderly, refuting the risk myth (Paterson & Warburton, 2010; Chodzko-Zajko et al., 2009). |- | Positive thinking is naive. || Adopting a positive mindset is empirically supported, and positive psychology boosts well-being in the elderly (Seligman et al., 2005; Sin & Lyubomirsky, 2009). |} == Significance of psychological flourishing for the elderly == Psychological flourishing holds a special significance for the elderly (Seligman, 2011). As they navigate the complexities of ageing, fostering a sense of purpose, joy, and well-being is essential (Bonanno et al., 2004). The importance of psychological flourishing for older adults lies in its comprehensive contributions to their emotional, cognitive, and physical domains (Emmons & McCullough, 2003; Hultsch et al., 1999). Drawing upon current research, the following discussion explores the transformative effects of a flourishing mindset. === Emotional benefits === Ageing introduces challenges that can be emotionally strenuous. However, research such as that by Peterson et al. (2007) indicates that character strengths, including love, gratitude, and hope, can significantly enhance emotional well-being. A good sense of humour provides further enhancement to emotional well-being according to research conducted by Martin et al. (1993). Flourishing ensures that the elderly possess the emotional resilience required to confront challenges, nurturing feelings of contentment and fulfilment (Bonanno et al., 2004). A senior who flourishes, for instance, might derive profound joy from meaningful relationships or experience a deep sense of gratitude for life's journey. This emotional equilibrium not only improves overall life quality but also acts as a safeguard against stress, rendering the ageing experience more rewarding (Bengtson, 2001; Carstensen et al., 1999). === Cognitive benefits === Psychological flourishing has discernible effects on cognitive function. A flourishing mind is an active one, continuously engaged in stimulating activities. Various researchers have observed that aspects like mental stimulation and engagement in activities such as walking, cycling, and sports play a role in cognitive maintenance (Bonanno et al., 2004; Emmons & McCullough, 2003; Hultsch et al., 1999). Cross-sectional and retrospective studies, although lacking direct causation evidence, highlight the correlation between physical activity and cognitive function (Hultsch et al., 1999; Paterson & Warburton, 2010). Moreover, character strengths such as curiosity and zest, as pinpointed by Peterson et al. (2007), equip the elderly with a passion for learning, keeping their cognitive faculties sharp and agile. It is a proactive stance against cognitive decline, ensuring that the elderly remain mentally active and engaged. === Impact on physical health and longevity === The connection between the mind and body is profound, particularly in the context of ageing. Research consistently underscores the positive effects of physical activity on functional outcomes, with an emphasis on the benefits of aerobic activities and structured exercise programs (Greenfield & Marks, 2004; Paterson & Warburton, 2010). Notably, regular participation in these activities is linked to decreased risks of functional impairments. Beyond the immediate physical benefits, psychological flourishing, with its focus on positive behaviours and proactive approaches, may also enhance longevity (Bengtson, 2001; Peterson et al. 2007). For example, seniors who cultivate qualities such as gratitude, as identified in studies by Peterson et al. (2007), are likely to adopt more health-promoting behaviours. This symbiotic relationship between a flourishing mind and a healthy body underscores the importance of psychological well-being in the elderly. {{RoundBoxTop|theme=6}} <quiz display=simple> {Imagine you are chatting with a friend about the advantages of psychological flourishing in older adults. Which of the following is NOT a benefit of psychological flourishing in the elderly? |type="()"} - Enhanced emotional resilience to confront challenges + Safeguard against financial issues - Improved overall life quality - Acts as a safeguard against stress {A senior with a flourishing mindset might derive profound joy from meaningful relationships and experience a deep sense of gratitude for life's journey. |type="()"} + True - False {Research indicates that seniors who cultivate qualities like gratitude are more likely to: |type="()"} - Avoid social interactions - Abstain from any form of physical activity + Adopt additional health-promoting behaviours - Experience rapid cognitive decline </quiz> {{RoundBoxBottom}} == The role of positive psychology == Positive psychology represents a rapidly growing field, concentrating on the cultivation of individual strengths, virtues, and peak human functioning, as highlighted by Seligman (2011). For elderly individuals, this approach provides a comprehensive viewpoint, tackling the intrinsic challenges associated with ageing while actively encouraging a flourishing life that extends past mere existence. Southwick and Charney's (2012) influential research underscores resilience, an essential aspect of positive psychology's resilience theory, as crucial in managing stress and trauma. This section explores relevant core principles of positive psychology that, illustrate its substantial utility in supporting psychological flourishing among seniors. === Foundational principles and theories === Positive psychology fundamentally examines the conditions and processes essential for optimal functioning, as investigated by Southwick and Charney (2012). It transforms the traditional focus on deficits and disorders, advocating instead for the amplification of strengths, positive emotions, and resilience. These elements are crucial for seniors, offering invaluable guidance through the complexities of later life. Core theoretical frameworks within this field, including resilience theory, self-determination theory, and well-being theory, shed light on various avenues available for seniors to discover meaning, purpose, and an elevated sense of well-being, even amidst the challenges life presents. * '''Resilience theory''' in psychology delves into the complex processes that enable individuals to adapt positively in the face of adversity. Recognising resilience as a dynamic and developmental capacity, it encompasses both internal cognitive-affective mechanisms, such as emotion regulation and a robust sense of self-efficacy, and external factors like strong social support networks. This framework underscores the notion that resilience is not an innate trait; rather, it is a skill subject to cultivation. The theory also integrates the concept of post-traumatic growth, shedding light on how challenging experiences can contribute to personal development and a deeper sense of meaning in life. By considering individual differences and the developmental context, resilience theory provides a nuanced understanding of adaptive functioning, guiding interventions that aim to enhance resilience and promote positive psychological outcomes (Southwick & Charney, 2012; Bonanno et al., 2004). * '''Self-determination theory''' (SDT) stands as a prominent psychological framework elucidating the intrinsic and extrinsic factors driving human motivation and behaviour. Developed by Deci and Ryan, the theory posits that the fulfillment of three core psychological needs—autonomy, competence, and relatedness—is paramount for optimal functioning and well-being (Ryan & Deci, 2018). Autonomy pertains to the sense of volition and self-governance in one’s actions, competence encompasses the need to master tasks and learn, and relatedness refers to the desire for meaningful connections and belongingness. In the context of flourishing, particularly among seniors, SDT offers valuable insights into how supportive environments and interventions can be crafted to nurture these fundamental needs. By emphasising the integral role of internal motivation and the social context in psychological well-being, SDT provides a nuanced and comprehensive understanding of the pathways leading to enhanced flourishing and life satisfaction. This theory, therefore, serves as a pivotal guide in the quest to foster resilient, connected, and self-determined lives in the elderly population. * '''Well-being theory''', conceptualised by Martin Seligman, represents a pivotal shift in psychological thought, extending the focus from the mere alleviation of distress to the active cultivation of optimal living. Seligman (2011) introduced the PERMA model, which encapsulates five integral components of well-being: positive emotion, engagement, positive relationships, meaning, and accomplishment. Positive emotion emphasises the value of fostering joy and contentment, while engagement stresses the importance of deep absorption in activities, invoking a state of flow. Positive relationships highlight the necessity of nurturing supportive and enriching connections. Meaning pertains to pursuing a purpose larger than oneself, and accomplishment encompasses striving for mastery and achievement. For the elderly, these dimensions provide a comprehensive blueprint for flourishing, guiding interventions and practices aimed at enhancing life quality and fulfilment. Well-being theory thus stands as a foundational pillar in positive psychology, offering a nuanced and holistic understanding of the factors that contribute to a life well-lived, irrespective of one’s age. === Psychological functioning and its role === Research conducted by Southwick and Charney (2012) underscores the integral role of psychological resilience in combatting depression—a condition prevalent among the elderly (16.1% of 65+-year-old Australians in [https://www.abs.gov.au/statistics/health/health-conditions-and-risks/health-conditions-prevalence/latest-release 2020–21]). Resilience does not merely denote bouncing back from adversity but signifies thriving amidst it. The elderly, with a lifetime of experiences, possess unique strengths and coping mechanisms. By harnessing these, and understanding neurobiological and psychosocial factors, there is an avenue for improved psychological functioning, ultimately contributing to enhanced flourishing. === The resilience connection === Several studies highlight that resilience is intricately linked to genetics, environment, neurobiology, and psychosocial factors (Helliwell & Sachs, 2013; Seligman, 2011; Southwick & Charney, 2012). For seniors, building resilience becomes a pivotal tool in managing age-related stressors and traumas. Embracing resilience does not mean avoiding challenges but rather developing a robust toolkit to face them head-on. With strategies encompassing positive emotions, social support, coping skills, and more, resilience offers a protective layer, allowing seniors to remain buoyant in the face of life's trials. === Proactive approaches and interventions === Promoting flourishing among the elderly necessitates actionable steps grounded in empirical evidence. Southwick and Charney (2012) advocate for interventions such as modifying the biological and psychosocial environment, strengthening social support networks, enhancing cognitive engagements, and boosting physical health through aerobic exercises. It is crucial to tailor these approaches to the distinct needs of seniors, taking into account their physiological, cognitive, and socio-emotional states. When informed by positive psychology, the amalgamation of these interventions paves the way for a life marked by heightened contentment, purpose, and joy in the latter years. Figure 3 illustrates how the integration of these supportive interventions correlates with increased levels of psychological flourishing among the elderly.<br> [[File:Psychological Flourishing - amalgamation model.png|center|thumb|400x400px|'''Figure 3'''. ''The cumulative impact of multiple interventions for supporting flourishing in the elderly''.]] == What can seniors do to flourish? == Psychological flourishing in seniors is essential for their well-being and quality of life. This section delves into practical, evidence-based strategies derived from positive psychology to foster flourishing among the elderly. It emphasises the importance of social engagement, mental stimulation, physical activity, and positive coping techniques. Tailored to individual needs, these approaches aim to enhance purpose, joy, and meaningful connections, addressing the unique challenges faced by seniors in their pursuit of a fulfilling life. === Social engagement and meaningful activities === * Building social connections rejuvenates the spirit and enhances well-being. * Social relationships have been found to significantly influence mental and emotional well-being in the elderly. A strong social network can reduce feelings of loneliness and depression, thereby promoting psychological flourishing (Cacioppo et al., 2006). * Forming bonds across different age groups can be mutually beneficial and specifically aid the elderly in feeling more connected and less isolated (Bengtson, 2001). * Engaging in community activities or volunteering (see Figure 4) has been shown to provide a sense of purpose and improve mental health outcomes for seniors (Greenfield & Marks, 2004). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': When was the last time you made a new friend? What activities can help you meet new people? :''Suggestions'': Engage in community activities, reconnect with old friends, or even consider pet ownership.<br> ::Establish regular family visits or calls, attend local gatherings, or join clubs focused on specific interests. |} [[File:Deeply engrossed in puzzle.png|thumb|317x317px|'''Figure 5'''''.'' ''An elderly woman deeply engrossed in her daily crossword puzzle; an excellent form of mental stimulation''.]] === Mental stimulation === * Continuous learning has been linked to cognitive vitality and emotional well-being (Park & Bischof, 2013). * Activities that require creativity, such as painting or music, not only stimulate the brain but also contribute to a greater sense of purpose and joy, enhancing the quality of life (Cohen et al., 2006). * Research indicates that lifelong learning and mental stimulation can help prevent cognitive decline and improve overall psychological well-being. Older adults who engage in mentally stimulating activities (see Figure 5) report higher levels of happiness and lower levels of depression (Hultsch et al., 1999). * More recent research underscores the notion that the ageing brain is capable of new neural connections when subjected to novel learning experiences, enhancing cognitive and emotional well-being (Park & Bischof, 2013). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Take up a new hobby, join an interesting class, or simply read a new book.<br> ::Explore online courses tailored for seniors or consider group-based activities to foster intellectual engagement. |} === Physical activity === [[File:TaiChi-group.png|thumb|300x280px|'''Figure 6'''''.'' ''A group of elderly enjoying a tai chi class tailored for maintaining mobility and easing arthritis symptoms''.]] * Regular physical activity boosts mental health and protects against age-related ailments such as heart disease, osteoporosis, and [https://en.wikipedia.org/wiki/Sarcopenia sarcopenia] (loss of muscle mass). * Several studies show that regular physical exercise can improve cognitive function, thereby supporting not only physical but also mental well-being (Colcombe & Kramer, 2003). * Group exercise activities like tai chi (see Figure 6) or water aerobics offer not only physical benefits but also social interaction, which can further contribute to psychological well-being (Liu & Latham, 2009). * Improved sleep through regular physical activity is correlated with better mood and mental health, providing another pathway to psychological flourishing (Reid et al., 2010). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Incorporate simple exercises into your daily routine, such as ::walking, yoga, or dancing, or consider joining a senior-friendly exercise group.<br> :''Please note:'' Always consult your healthcare professional before starting any new exercise regimen. |} [[File:Meditating man-solo.png|thumb|317x317px|'''Figure 7'''''.'' ''An elderly man practicing guided meditation; an excellent positive coping strategy''.]] === Positive coping strategies === * Develop resilience against challenges by adopting positive coping mechanisms such as meditation (see Figure 7), relaxation techniques, and engaging in spirituality. * Positive psychology interventions focus on strengths and virtues and have shown efficacy in improving well-being and reducing depressive symptoms in older adults (Seligman et al., 2005). * Incorporating gratitude into daily routines has been associated with positive emotional states, greater well-being, and better physical health in older adults (Emmons & McCullough, 2003). * Utilising humor is shown to not only uplift mood but also serve as an effective coping strategy for stress and life challenges. This is particularly relevant for elderly individuals who may face various forms of age-related adversity (Martin et al., 1993). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a past challenge. How did you cope? How can you refine this strategy? :''Suggestions'': Embrace activities that foster mindfulness and reflection, such as ::meditation, gratitude journaling, or joining a support group.<br> |} {{RoundBoxTop|theme=13}} [[File:Elderly man-zest for life-mentoring.png|thumb|327x182px|'''Figure 8'''. ''John rediscovered his zest for life through mentoring teenagers, a program organised by his local APEX club''.]] '''''Can you relate''''' ... John, at 80, often sat in his armchair, lost in memories of youthful adventures. He believed his golden years had long passed, with each day echoing the sentiments of a vibrant past. Collecting his mail one day, a flyer for the local book club caught his attention, hinting at a promise of engaging discussions. Deciding to join, John discovered that the club was more than just about books; it was a community bridging generational gaps through shared stories. A young woman from the group, impressed by John's vast life experiences, introduced him to a mentoring program hosted by the local [[w:Apex Clubs of Australia|APEX]] club. Through mentoring, John shared his life lessons, offering wisdom and guidance to the younger generation. This exchange rekindled his understanding of the value of his own journey. Far from feeling that his best years were behind him, the interactions brought about a renewed sense of purpose (see Figure 8). Through the book club and mentoring, John not only found a revived passion for literature but also tapped into a deeper zest for life, seeing his age not as a limitation but as a testament to a life rich with experiences. {{RoundBoxBottom}} ==Conclusion== Ageing, despite its inherent challenges, can still be a period rich in growth, connection, and profound meaning. By integrating the principles of positive psychology with tailored interventions, an environment conducive to psychological flourishing in the elderly can be cultivated. Foundational theories such as resilience theory, self-determination theory, and well-being theory offer a robust framework for understanding and nurturing the various facets of flourishing. Translating these theories into practical strategies and interventions empowers seniors to gracefully navigate the complexities of ageing, maintaining resilience and a sustained sense of well-being. Supporting flourishing in the elderly necessitates a commitment to social engagement, enhancing emotional well-being; continuous mental stimulation, promoting cognitive vitality; regular physical activity, supporting both physical and mental health; and the adoption of positive coping strategies to foster resilience. These strategies not only enhance their quality of life but also contribute positively to the broader community, as the elderly impart their wisdom, experience, and emotional stability. Furthermore, dispelling misconceptions surrounding ageing and psychological flourishing is paramount. Challenging societal stereotypes and adopting a holistic view of ageing is essential, acknowledging the potential for growth, development, and fulfilment in this life stage. In conclusion, supporting psychological flourishing in the elderly is a multifaceted endeavour. It demands a collaborative effort from individuals, communities, and society at large, aiming to create environments that nurture the emotional, cognitive, and physical aspects of well-being. In doing so, the invaluable contributions of the elderly are honoured, fostering a culture of respect, appreciation, and comprehensive support, ensuring that the senior years are indeed filled with growth, connection, and a profound sense of purpose. ==See also== # [[Motivation and emotion/Book/2014/Ageing and emotion|Ageing and emotion]] (Book chapter, 2014) # [[Motivation and emotion/Book/2023/Ageing and motivation|Ageing and motivation]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Nudge_theory_and_sedentary_behaviour|Nudge theory and sedentary behaviour]] (Book chapter, 2023) # [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) # [[Motivation and emotion/Book/2023/Death_and_meaning_in_life|Death and meaning in life]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Community_resilience|Community resilience]] (Book chapter, 2023) == References == {{Hanging indent|1= Ball, K., Berch, D., Helmers, K., Jobe, J., Leveck, M., Marsiske, M., Morris, J., Rebok, G., Smith, D., Tennstedt, S., Unverzagt, F., & Willis, S. (2002). Effects of cognitive training interventions with older adults. ''JAMA'', ''288''(18), 2271–2281. https://doi.org/10.1001/jama.288.18.2271 Bengtson, V. L. (2001). Beyond the nuclear family: The increasing importance of multigenerational bonds. ''Journal of Marriage and Family'', ''63''(1), 1–16. https://doi.org/10.1111/j.1741-3737.2001.00001.x Bonanno, G. A., Wortman, C. B., & Nesse, R. M. (2004). Prospective patterns of resilience and maladjustment during widowhood. ''Psychology and Aging'', ''19''(2), 260–271. https://doi.org/10.1037/0882-7974.19.2.260 Cacioppo, J. T., Hughes, M. E., Waite, L. J., Hawkley, L. C., & Thisted, R. A. (2006). Loneliness as a specific risk factor for depressive symptoms: Cross-sectional and longitudinal analyses. ''Psychology and Aging'', ''21''(1), 140–151. https://doi.org/10.1037/0882-7974.21.1.140 Carstensen, L. L., Isaacowitz, D. M., & Charles, S. T. (1999). Taking time seriously: A theory of socioemotional selectivity. ''American Psychologist'', ''54''(3), 165–181. https://doi.org/10.1037/0003-066X.54.3.165 Charles, S. T., & Carstensen, L. L. (2010). Social and emotional aging. ''Annual Review of Psychology'', ''61'', 383–409. https://doi.org/10.1146/annurev.psych.093008.100448 Chodzko-Zajko, W. J., Proctor, D. N., Fiatarone Singh, M. A., Minson, C. T., Nigg, C. R., Salem, G. J., & Skinner, J. S. (2009). Exercise and physical activity for older adults. ''Medicine & Science in Sports & Exercise'', ''41''(7), 1510–1530. https://doi.org/10.1249/MSS.0b013e3181a0c95c Cohen, G. D., Perlstein, S., Chapline, J., Kelly, J., Firth, K. M., & Simmens, S. (2006). The impact of professionally conducted cultural programs on the physical health, mental health, and social functioning of older adults. ''The Gerontologist'', ''46''(6), 726–734. https://doi.org/10.1093/geront/46.6.726 Colcombe, S., & Kramer, A. F. (2003). Fitness effects on the cognitive function of older adults: A meta-analytic study. ''Psychological Science'', ''14''(2), 125–130. https://doi.org/10.1111/1467-9280.t01-1-01430 Emmons, R. A., & McCullough, M. E. (2003). Counting blessings versus burdens: An experimental investigation of gratitude and subjective well-being in daily life. ''Journal of Personality and Social Psychology'', ''84''(2), 377–389. https://doi.org/10.1037/0022-3514.84.2.377 Greenfield, E. A., & Marks, N. F. (2004). Formal volunteering as a protective factor for older adults' psychological well-being. ''The Journals of Gerontology Series B: Psychological Sciences and Social Sciences'', ''59''(5), S258–S264. https://doi.org/10.1093/geronb/59.5.S258 Helliwell, J. F., Layard, R., & Sachs, J. (2013). World happiness report. ''United Nations Sustainable Development Solutions Network''. http://eprints.lse.ac.uk/47487/ Hultsch, D. F., Hertzog, C., Small, B. J., & Dixon, R. A. (1999). Use it or lose it: Engaged lifestyle as a buffer of cognitive decline in aging? ''Psychology and Aging'', ''14''(2), 245–263. https://doi.org/10.1037/0882-7974.14.2.245 Keyes, C. L. M. (2007). Promoting and protecting mental health as flourishing: A complementary strategy for improving national mental health. ''American Psychologist'', ''62''(2), 95–108. https://doi.org/10.1037/0003-066X.62.2.95 Liu, C. J., & Latham, N. K. (2009). Progressive resistance strength training for improving physical function in older adults. ''The Cochrane Database of Systematic Reviews'', (3), CD002759. https://doi.org/10.1002/14651858.CD002759.pub2 Martin, R. A., Kuiper, N. A., Olinger, L. J., & Dance, K. A. (1993). Humor, coping with stress, self-concept, and psychological well-being. ''Humor-International Journal of Humor Research'', ''6''(1), 89–104. https://doi.org/10.1515/humr.1993.6.1.89 Park, D. C., & Bischof, G. N. (2013). The aging mind: neuroplasticity in response to cognitive training. ''Dialogues in Clinical Neuroscience'', ''15''(1), 109–119. https://doi.org/10.31887/DCNS.2013.15.1/dpark Park, D. C., Lautenschlager, G., Hedden, T., Davidson, N. S., Smith, A. D., & Smith, P. K. (2002). Models of visuospatial and verbal memory across the adult life span. ''Psychology and Aging'', ''17''(2), 299–320. https://doi.org/10.1037/0882-7974.17.2.299 Paterson, D. H., & Warburton, D. E. (2010). Physical activity and functional limitations in older adults: a systematic review related to Canada's physical activity guidelines. ''International Journal of Behavioral Nutrition and Physical Activity'', ''7''(1), 1–22. https://doi.org/10.1186/1479-5868-7-38 Peterson, C., Ruch, W., Beermann, U., Park, N., & Seligman, M. E. P. (2007). Strengths of character, orientations to happiness, and life satisfaction. ''The Journal of Positive Psychology'', ''2''(3) 149–156. https://doi.org/10.1080/17439760701228938 Reid, K. J., Baron, K. G., Lu, B., Naylor, E., Wolfe, L., & Zee, P. C. (2010). Aerobic exercise improves self-reported sleep and quality of life in older adults with insomnia. ''Sleep Medicine'', ''11''(9), 934–940. https://doi.org/10.1016/j.sleep.2010.04.014 Ryan, R. M., & Deci, E. L. (2018). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. The Guildford Press. Seligman, M. E. P. (2011). ''Flourish''. William Heinnemann. Seligman, M. E. P., Steen, T. A., Park, N., & Peterson, C. (2005). Positive psychology progress: Empirical validation of interventions. ''American Psychologist'', ''60''(5), 410. https://doi.org/10.1037/0003-066X.60.5.410 Sin, N. L., & Lyubomirsky, S. (2009). Enhancing well-being and alleviating depressive symptoms with positive psychology interventions: A practice-friendly meta-analysis. ''Journal of Clinical Psychology'', ''65''(5), 467–487. https://doi.org/10.1002/jclp.20593 Southwick, S. M., & Charney, D. S. (2012) The science of resilience: Implications for the prevention and treatment of depression. ''Science'', ''338''(6103), 79–82. https://doi.org/10.1126/science.1222942 }} == External links == # [https://youtu.be/Is6WvYAM3gg?si=KWHkkgmP47QQDf0O 100-year olds' guide to living your best life] (Allure; YouTube) # [https://www.bluezones.com/2016/11/power-9/ Blue zones power 9: Lifestyle habits of the world’s healthiest, longest-lived people] (bluezones.com) # [https://ppc.sas.upenn.edu/ Positive psychology center] (University of Pennsylvania) # [https://youtu.be/bPBJJ-lxsXA?si=f2yDgG9X6lqiw4Cp The secret to successful aging] (Cathleen Toomey; TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:{{#titleparts:{{PAGENAME}}|3}}/Top]] [[Category:Motivation and emotion/Book/Ageing]] [[Category:Motivation and emotion/Book/Flourishing]] gy0kql5m9tl5ei6thtgekvylzcbk0t3 2803456 2803454 2026-04-08T02:45:09Z Jtneill 10242 /* Overview */ + image for Figure 1 2803456 wikitext text/x-wiki {{title|Flourishing in the elderly:<br>How can psychological flourishing be supported in the elderly?}} {{MECR3|1=https://youtu.be/yUAuPhv5S4o}} __TOC__ ==Overview== <br> {{RoundBoxTop|theme=13}} [[File:Elderly Woman, B&W image by Chalmers Butterfield.jpg|thumb|157x196px|'''Figure 1'''. ''Sarah ...''. (image no longer visible after April 2025 due to Wiki Commons feedback and user request for deletion)]] '''''Can you relate''''' ... Sarah, aged 85, begins her day in a house echoing with memories. Faded photographs on the mantelpiece showcase her once pivotal role in the community. From hosting gatherings to being an active voice at the local council, she had always championed causes dear to her. But now, an overwhelming quietness envelops her heart, and she has lost touch with her sense of purpose. Time has seen her closest friends move or pass away. Family visits, once frequent and filled with laughter, have become increasingly rare. This growing isolation weighs on Sarah, exacerbated by a society that seems to prioritise youth over experience. The fast-paced technological world further alienates her; smart devices and social media platforms feel foreign, exacerbating her deep-seated fear of irrelevance in a world that is rapidly evolving without her. In a society increasingly centred on youth, how can seniors like Sarah reclaim their vitality and sense of meaning? {{RoundBoxBottom}} <br> [[File:Optic_stub.svg|thumb|250x250px|'''Figure 2'''. ''An elderly couple ...'' (image no longer visible after March 2026 due to Wiki Commons feedback and user request for deletion)]] The [[wikipedia:Ageing#Successful_ageing|ageing]] population represents an invaluable repository of [[wikipedia:Wisdom|wisdom]], [[wikipedia:Experience|experience]], and [[wikipedia:Insight|insight]] (see Figure 2). Yet, many elderly individuals, much like Sarah, face challenges related to purpose, meaning, and overall psychological [[wikipedia:Well-being|well-being]]. This raises a question; can the elderly attain a state of psychological [[Motivation and emotion/Book/2018/Flourishing|flourishing]], even in the face of age-related adversities? The domain of [[wikipedia:Positive_psychology|positive psychology]] offers evidence-based strategies. Read on to explore what these strategies are and how they can be applied to support seniors in their journey toward a life filled with purpose, meaning, and joy. Specifically, psychological flourishing in the elderly may be supported through a combination of the following four elements, especially when tailored to individual needs and preferences: # [[wikipedia:Social_support|Social engagement]] and meaningful activities such as community involvement or [[wikipedia:Hobby#Psychological_role|hobbies]] (Helliwell et al., 2013) # Mental stimulation via [[wikipedia:Brain_training|cognitive training]] programs can sustain mental acuity and may delay cognitive decline (Ball et al., 2002) # [[Motivation and emotion/Book/2013/Motivating the elderly to exercise|Physical activity]] using exercises tailored to the elderly improve mood and cognitive function (Colcombe & Kramer, 2003) # Incorporating [[Motivation and emotion/Book/2020/Coping and emotion|positive coping strategies]] from positive psychology techniques, such as gratitude exercises, builds [[wikipedia:Psychological_resilience|resilience]] and satisfaction (Seligman et al., 2005) <br> {{RoundBoxTop|theme=3}} [[File:Optics stub.png|right|60px|]] '''Focus questions:''' * What is psychological flourishing? * Why is psychological flourishing important for the elderly? * How does positive psychology foster psychological flourishing? * What can seniors do to flourish? {{RoundBoxBottom}} <br> == What is psychological flourishing? == Psychological flourishing, a term frequently linked to positive psychology, has gained prominence over the years. For elderly individuals, understanding and achieving psychological flourishing is paramount, considering the myriad challenges they confront during this stage of life (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theory, such as self-determination theory and resilience theory, combine with empirical evidence to underpin the suggested four-element model. It is also essential to define and distinguish psychological flourishing from broader notions of general flourishing and well-being. Through understanding this distinction, the unique facets of flourishing and its potential impact on the lives of older individuals become evident. Common misconceptions about psychological flourishing are also addressed, ensuring clarity in subsequent discussions. Psychological flourishing, a concept closely associated with positive psychology, has gained prominence in recent years. For elderly individuals, comprehending and attaining psychological flourishing is crucial, given the diverse challenges encountered in this life stage (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theories such as self-determination theory and resilience theory, as well as empirical evidence, will be used to establish a four-element model of psychological flourishing. Additionally, it is vital to delineate psychological flourishing from broader concepts of general flourishing and well-being. By grasping these distinctions, the unique facets of flourishing and their potential transformative impacts on the lives of older individuals become evident. Common misconceptions about psychological flourishing will also be addressed to ensure clarity in the discussions that follow. === Definitions and distinctions === Psychological flourishing, often simply termed "flourishing", signifies a pinnacle in human functioning. It represents a well-being state that extends beyond merely being free from distress or psychological issues. More than mere survival, flourishing connotes thriving, excelling, and feeling an intense sense of purpose and contentment (Seligman et al., 2005). This condition transcends mere happiness or the absence of mental illness (Sin & Lyubomirsky, 2009). Flourishing provides a holistic view, emphasising positive human functioning across various areas, from relationships to personal growth and purpose. It captures positive emotions, a sense of engagement, strong social connections, and a profound understanding of life's meaning. Distinct from general well-being, flourishing underscores not just feeling good but also functioning effectively, accentuating both [[wikipedia:Hedonic_motivation|hedonic]] (feelings about life) and [[wikipedia:Eudaimonia|eudaimonic]] (functioning in life) well-being facets (Keyes, 2007). Positive psychology’s well-being theory, also known as the PERMA model, conceptualises well-being as a state of comfort, health, or happiness (Seligman, 2011). The model comprises five elements: positive emotion, engagement, positive relationships, meaning, and accomplishment (Seligman, 2011). However, psychological flourishing transcends the ‘feeling good’ aspect of well-being. It encompasses optimal functioning in daily life, incorporating the key principles of self-determination theory: autonomy, relatedness, and competence. === Importance in the context of ageing === Ageing often elicits feelings of apprehension and resignation, influenced by perceptions of decline or limitation, both physically and mentally. Nevertheless, research indicates that many elderly individuals experience periods of growth, insight, and enhanced well-being. For example, the Adult Development and Enrichment Project (ADEPT), conducted by researchers at Pennsylvania State University, underscored the potential for intellectual and emotional growth even in advanced age (Hultsch et al., 1999). Psychological flourishing becomes paramount in this context, assisting the elderly in navigating challenges, capitalising on opportunities, and making meaningful contributions to their communities and personal lives, thereby maintaining their autonomy, relatedness, and competence. Psychological flourishing challenges these notions of decline or limitation, highlighting the profound possibilities for growth, enrichment, and depth in later life (Paterson & Warburton, 2010). Embracing flourishing can significantly transform the way elderly individuals perceive their later years, encouraging them to view this period as an opportunity for renewed purpose, deepened relationships, and the cultivation of new passions or interests. Furthermore, as elderly individuals inevitably face challenges, resilience theory provides a framework for developing a mindset rooted in flourishing (Southwick & Charney, 2012; Bonanno et al., 2004). Such a capacity and mindset act as a solid foundation, empowering seniors to approach obstacles with resilience and poise (Seligman et al., 2005). === Misconceptions === Several misconceptions exist regarding the concept of psychological flourishing. Contrary to common belief, it does not imply a perpetual state of happiness or a life devoid of adversity. Psychological flourishing is not about the absence of negative emotions or challenges. Instead, it focuses on cultivating psychological tools and strategies to navigate adversity, empowering individuals to thrive amidst difficulties (Carstensen et al., 1999). Clarifying these misconceptions paves the way for a deeper understanding of how older individuals can achieve and maintain psychological flourishing. To truly grasp psychological flourishing in older adults, it is crucial to debunk prevalent myths about the ageing process. If left unchallenged, these misconceptions could deter the implementation of strategies promoting well-being in later life (Carstensen et al., 1999; Charles & Carstensen, 2010). Table 1 presents evidence-based counters to these myths, highlighting the potential for growth and vitality among the elderly. {| class="wikitable" |+ Table 1. ''Dispelling Common Misconceptions about the Ageing Process''<br> |- ! Myth !! Fact |- | Ageing leads to inevitable cognitive and emotional decline. || Ageing can offer growth opportunities with the right strategies. |- | Ageing means mental and physical decline. || Many older adults maintain high cognitive and physical activity levels with suitable exercises (Park et al., 2002; Hultsch et al., 1999). |- | Social withdrawal is an ageing inevitability. || While social circles might decrease in size, relationship quality often improves, and staying socially engaged is beneficial (Carstensen et al., 1999; Charles & Carstensen, 2010). |- | Physical activity is risky for older adults. || Moderate physical activity improves mental and physical health in the elderly, refuting the risk myth (Paterson & Warburton, 2010; Chodzko-Zajko et al., 2009). |- | Positive thinking is naive. || Adopting a positive mindset is empirically supported, and positive psychology boosts well-being in the elderly (Seligman et al., 2005; Sin & Lyubomirsky, 2009). |} == Significance of psychological flourishing for the elderly == Psychological flourishing holds a special significance for the elderly (Seligman, 2011). As they navigate the complexities of ageing, fostering a sense of purpose, joy, and well-being is essential (Bonanno et al., 2004). The importance of psychological flourishing for older adults lies in its comprehensive contributions to their emotional, cognitive, and physical domains (Emmons & McCullough, 2003; Hultsch et al., 1999). Drawing upon current research, the following discussion explores the transformative effects of a flourishing mindset. === Emotional benefits === Ageing introduces challenges that can be emotionally strenuous. However, research such as that by Peterson et al. (2007) indicates that character strengths, including love, gratitude, and hope, can significantly enhance emotional well-being. A good sense of humour provides further enhancement to emotional well-being according to research conducted by Martin et al. (1993). Flourishing ensures that the elderly possess the emotional resilience required to confront challenges, nurturing feelings of contentment and fulfilment (Bonanno et al., 2004). A senior who flourishes, for instance, might derive profound joy from meaningful relationships or experience a deep sense of gratitude for life's journey. This emotional equilibrium not only improves overall life quality but also acts as a safeguard against stress, rendering the ageing experience more rewarding (Bengtson, 2001; Carstensen et al., 1999). === Cognitive benefits === Psychological flourishing has discernible effects on cognitive function. A flourishing mind is an active one, continuously engaged in stimulating activities. Various researchers have observed that aspects like mental stimulation and engagement in activities such as walking, cycling, and sports play a role in cognitive maintenance (Bonanno et al., 2004; Emmons & McCullough, 2003; Hultsch et al., 1999). Cross-sectional and retrospective studies, although lacking direct causation evidence, highlight the correlation between physical activity and cognitive function (Hultsch et al., 1999; Paterson & Warburton, 2010). Moreover, character strengths such as curiosity and zest, as pinpointed by Peterson et al. (2007), equip the elderly with a passion for learning, keeping their cognitive faculties sharp and agile. It is a proactive stance against cognitive decline, ensuring that the elderly remain mentally active and engaged. === Impact on physical health and longevity === The connection between the mind and body is profound, particularly in the context of ageing. Research consistently underscores the positive effects of physical activity on functional outcomes, with an emphasis on the benefits of aerobic activities and structured exercise programs (Greenfield & Marks, 2004; Paterson & Warburton, 2010). Notably, regular participation in these activities is linked to decreased risks of functional impairments. Beyond the immediate physical benefits, psychological flourishing, with its focus on positive behaviours and proactive approaches, may also enhance longevity (Bengtson, 2001; Peterson et al. 2007). For example, seniors who cultivate qualities such as gratitude, as identified in studies by Peterson et al. (2007), are likely to adopt more health-promoting behaviours. This symbiotic relationship between a flourishing mind and a healthy body underscores the importance of psychological well-being in the elderly. {{RoundBoxTop|theme=6}} <quiz display=simple> {Imagine you are chatting with a friend about the advantages of psychological flourishing in older adults. Which of the following is NOT a benefit of psychological flourishing in the elderly? |type="()"} - Enhanced emotional resilience to confront challenges + Safeguard against financial issues - Improved overall life quality - Acts as a safeguard against stress {A senior with a flourishing mindset might derive profound joy from meaningful relationships and experience a deep sense of gratitude for life's journey. |type="()"} + True - False {Research indicates that seniors who cultivate qualities like gratitude are more likely to: |type="()"} - Avoid social interactions - Abstain from any form of physical activity + Adopt additional health-promoting behaviours - Experience rapid cognitive decline </quiz> {{RoundBoxBottom}} == The role of positive psychology == Positive psychology represents a rapidly growing field, concentrating on the cultivation of individual strengths, virtues, and peak human functioning, as highlighted by Seligman (2011). For elderly individuals, this approach provides a comprehensive viewpoint, tackling the intrinsic challenges associated with ageing while actively encouraging a flourishing life that extends past mere existence. Southwick and Charney's (2012) influential research underscores resilience, an essential aspect of positive psychology's resilience theory, as crucial in managing stress and trauma. This section explores relevant core principles of positive psychology that, illustrate its substantial utility in supporting psychological flourishing among seniors. === Foundational principles and theories === Positive psychology fundamentally examines the conditions and processes essential for optimal functioning, as investigated by Southwick and Charney (2012). It transforms the traditional focus on deficits and disorders, advocating instead for the amplification of strengths, positive emotions, and resilience. These elements are crucial for seniors, offering invaluable guidance through the complexities of later life. Core theoretical frameworks within this field, including resilience theory, self-determination theory, and well-being theory, shed light on various avenues available for seniors to discover meaning, purpose, and an elevated sense of well-being, even amidst the challenges life presents. * '''Resilience theory''' in psychology delves into the complex processes that enable individuals to adapt positively in the face of adversity. Recognising resilience as a dynamic and developmental capacity, it encompasses both internal cognitive-affective mechanisms, such as emotion regulation and a robust sense of self-efficacy, and external factors like strong social support networks. This framework underscores the notion that resilience is not an innate trait; rather, it is a skill subject to cultivation. The theory also integrates the concept of post-traumatic growth, shedding light on how challenging experiences can contribute to personal development and a deeper sense of meaning in life. By considering individual differences and the developmental context, resilience theory provides a nuanced understanding of adaptive functioning, guiding interventions that aim to enhance resilience and promote positive psychological outcomes (Southwick & Charney, 2012; Bonanno et al., 2004). * '''Self-determination theory''' (SDT) stands as a prominent psychological framework elucidating the intrinsic and extrinsic factors driving human motivation and behaviour. Developed by Deci and Ryan, the theory posits that the fulfillment of three core psychological needs—autonomy, competence, and relatedness—is paramount for optimal functioning and well-being (Ryan & Deci, 2018). Autonomy pertains to the sense of volition and self-governance in one’s actions, competence encompasses the need to master tasks and learn, and relatedness refers to the desire for meaningful connections and belongingness. In the context of flourishing, particularly among seniors, SDT offers valuable insights into how supportive environments and interventions can be crafted to nurture these fundamental needs. By emphasising the integral role of internal motivation and the social context in psychological well-being, SDT provides a nuanced and comprehensive understanding of the pathways leading to enhanced flourishing and life satisfaction. This theory, therefore, serves as a pivotal guide in the quest to foster resilient, connected, and self-determined lives in the elderly population. * '''Well-being theory''', conceptualised by Martin Seligman, represents a pivotal shift in psychological thought, extending the focus from the mere alleviation of distress to the active cultivation of optimal living. Seligman (2011) introduced the PERMA model, which encapsulates five integral components of well-being: positive emotion, engagement, positive relationships, meaning, and accomplishment. Positive emotion emphasises the value of fostering joy and contentment, while engagement stresses the importance of deep absorption in activities, invoking a state of flow. Positive relationships highlight the necessity of nurturing supportive and enriching connections. Meaning pertains to pursuing a purpose larger than oneself, and accomplishment encompasses striving for mastery and achievement. For the elderly, these dimensions provide a comprehensive blueprint for flourishing, guiding interventions and practices aimed at enhancing life quality and fulfilment. Well-being theory thus stands as a foundational pillar in positive psychology, offering a nuanced and holistic understanding of the factors that contribute to a life well-lived, irrespective of one’s age. === Psychological functioning and its role === Research conducted by Southwick and Charney (2012) underscores the integral role of psychological resilience in combatting depression—a condition prevalent among the elderly (16.1% of 65+-year-old Australians in [https://www.abs.gov.au/statistics/health/health-conditions-and-risks/health-conditions-prevalence/latest-release 2020–21]). Resilience does not merely denote bouncing back from adversity but signifies thriving amidst it. The elderly, with a lifetime of experiences, possess unique strengths and coping mechanisms. By harnessing these, and understanding neurobiological and psychosocial factors, there is an avenue for improved psychological functioning, ultimately contributing to enhanced flourishing. === The resilience connection === Several studies highlight that resilience is intricately linked to genetics, environment, neurobiology, and psychosocial factors (Helliwell & Sachs, 2013; Seligman, 2011; Southwick & Charney, 2012). For seniors, building resilience becomes a pivotal tool in managing age-related stressors and traumas. Embracing resilience does not mean avoiding challenges but rather developing a robust toolkit to face them head-on. With strategies encompassing positive emotions, social support, coping skills, and more, resilience offers a protective layer, allowing seniors to remain buoyant in the face of life's trials. === Proactive approaches and interventions === Promoting flourishing among the elderly necessitates actionable steps grounded in empirical evidence. Southwick and Charney (2012) advocate for interventions such as modifying the biological and psychosocial environment, strengthening social support networks, enhancing cognitive engagements, and boosting physical health through aerobic exercises. It is crucial to tailor these approaches to the distinct needs of seniors, taking into account their physiological, cognitive, and socio-emotional states. When informed by positive psychology, the amalgamation of these interventions paves the way for a life marked by heightened contentment, purpose, and joy in the latter years. Figure 3 illustrates how the integration of these supportive interventions correlates with increased levels of psychological flourishing among the elderly.<br> [[File:Psychological Flourishing - amalgamation model.png|center|thumb|400x400px|'''Figure 3'''. ''The cumulative impact of multiple interventions for supporting flourishing in the elderly''.]] == What can seniors do to flourish? == Psychological flourishing in seniors is essential for their well-being and quality of life. This section delves into practical, evidence-based strategies derived from positive psychology to foster flourishing among the elderly. It emphasises the importance of social engagement, mental stimulation, physical activity, and positive coping techniques. Tailored to individual needs, these approaches aim to enhance purpose, joy, and meaningful connections, addressing the unique challenges faced by seniors in their pursuit of a fulfilling life. === Social engagement and meaningful activities === * Building social connections rejuvenates the spirit and enhances well-being. * Social relationships have been found to significantly influence mental and emotional well-being in the elderly. A strong social network can reduce feelings of loneliness and depression, thereby promoting psychological flourishing (Cacioppo et al., 2006). * Forming bonds across different age groups can be mutually beneficial and specifically aid the elderly in feeling more connected and less isolated (Bengtson, 2001). * Engaging in community activities or volunteering (see Figure 4) has been shown to provide a sense of purpose and improve mental health outcomes for seniors (Greenfield & Marks, 2004). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': When was the last time you made a new friend? What activities can help you meet new people? :''Suggestions'': Engage in community activities, reconnect with old friends, or even consider pet ownership.<br> ::Establish regular family visits or calls, attend local gatherings, or join clubs focused on specific interests. |} [[File:Deeply engrossed in puzzle.png|thumb|317x317px|'''Figure 5'''''.'' ''An elderly woman deeply engrossed in her daily crossword puzzle; an excellent form of mental stimulation''.]] === Mental stimulation === * Continuous learning has been linked to cognitive vitality and emotional well-being (Park & Bischof, 2013). * Activities that require creativity, such as painting or music, not only stimulate the brain but also contribute to a greater sense of purpose and joy, enhancing the quality of life (Cohen et al., 2006). * Research indicates that lifelong learning and mental stimulation can help prevent cognitive decline and improve overall psychological well-being. Older adults who engage in mentally stimulating activities (see Figure 5) report higher levels of happiness and lower levels of depression (Hultsch et al., 1999). * More recent research underscores the notion that the ageing brain is capable of new neural connections when subjected to novel learning experiences, enhancing cognitive and emotional well-being (Park & Bischof, 2013). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Take up a new hobby, join an interesting class, or simply read a new book.<br> ::Explore online courses tailored for seniors or consider group-based activities to foster intellectual engagement. |} === Physical activity === [[File:TaiChi-group.png|thumb|300x280px|'''Figure 6'''''.'' ''A group of elderly enjoying a tai chi class tailored for maintaining mobility and easing arthritis symptoms''.]] * Regular physical activity boosts mental health and protects against age-related ailments such as heart disease, osteoporosis, and [https://en.wikipedia.org/wiki/Sarcopenia sarcopenia] (loss of muscle mass). * Several studies show that regular physical exercise can improve cognitive function, thereby supporting not only physical but also mental well-being (Colcombe & Kramer, 2003). * Group exercise activities like tai chi (see Figure 6) or water aerobics offer not only physical benefits but also social interaction, which can further contribute to psychological well-being (Liu & Latham, 2009). * Improved sleep through regular physical activity is correlated with better mood and mental health, providing another pathway to psychological flourishing (Reid et al., 2010). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Incorporate simple exercises into your daily routine, such as ::walking, yoga, or dancing, or consider joining a senior-friendly exercise group.<br> :''Please note:'' Always consult your healthcare professional before starting any new exercise regimen. |} [[File:Meditating man-solo.png|thumb|317x317px|'''Figure 7'''''.'' ''An elderly man practicing guided meditation; an excellent positive coping strategy''.]] === Positive coping strategies === * Develop resilience against challenges by adopting positive coping mechanisms such as meditation (see Figure 7), relaxation techniques, and engaging in spirituality. * Positive psychology interventions focus on strengths and virtues and have shown efficacy in improving well-being and reducing depressive symptoms in older adults (Seligman et al., 2005). * Incorporating gratitude into daily routines has been associated with positive emotional states, greater well-being, and better physical health in older adults (Emmons & McCullough, 2003). * Utilising humor is shown to not only uplift mood but also serve as an effective coping strategy for stress and life challenges. This is particularly relevant for elderly individuals who may face various forms of age-related adversity (Martin et al., 1993). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a past challenge. How did you cope? How can you refine this strategy? :''Suggestions'': Embrace activities that foster mindfulness and reflection, such as ::meditation, gratitude journaling, or joining a support group.<br> |} {{RoundBoxTop|theme=13}} [[File:Elderly man-zest for life-mentoring.png|thumb|327x182px|'''Figure 8'''. ''John rediscovered his zest for life through mentoring teenagers, a program organised by his local APEX club''.]] '''''Can you relate''''' ... John, at 80, often sat in his armchair, lost in memories of youthful adventures. He believed his golden years had long passed, with each day echoing the sentiments of a vibrant past. Collecting his mail one day, a flyer for the local book club caught his attention, hinting at a promise of engaging discussions. Deciding to join, John discovered that the club was more than just about books; it was a community bridging generational gaps through shared stories. A young woman from the group, impressed by John's vast life experiences, introduced him to a mentoring program hosted by the local [[w:Apex Clubs of Australia|APEX]] club. Through mentoring, John shared his life lessons, offering wisdom and guidance to the younger generation. This exchange rekindled his understanding of the value of his own journey. Far from feeling that his best years were behind him, the interactions brought about a renewed sense of purpose (see Figure 8). Through the book club and mentoring, John not only found a revived passion for literature but also tapped into a deeper zest for life, seeing his age not as a limitation but as a testament to a life rich with experiences. {{RoundBoxBottom}} ==Conclusion== Ageing, despite its inherent challenges, can still be a period rich in growth, connection, and profound meaning. By integrating the principles of positive psychology with tailored interventions, an environment conducive to psychological flourishing in the elderly can be cultivated. Foundational theories such as resilience theory, self-determination theory, and well-being theory offer a robust framework for understanding and nurturing the various facets of flourishing. Translating these theories into practical strategies and interventions empowers seniors to gracefully navigate the complexities of ageing, maintaining resilience and a sustained sense of well-being. Supporting flourishing in the elderly necessitates a commitment to social engagement, enhancing emotional well-being; continuous mental stimulation, promoting cognitive vitality; regular physical activity, supporting both physical and mental health; and the adoption of positive coping strategies to foster resilience. These strategies not only enhance their quality of life but also contribute positively to the broader community, as the elderly impart their wisdom, experience, and emotional stability. Furthermore, dispelling misconceptions surrounding ageing and psychological flourishing is paramount. Challenging societal stereotypes and adopting a holistic view of ageing is essential, acknowledging the potential for growth, development, and fulfilment in this life stage. In conclusion, supporting psychological flourishing in the elderly is a multifaceted endeavour. It demands a collaborative effort from individuals, communities, and society at large, aiming to create environments that nurture the emotional, cognitive, and physical aspects of well-being. In doing so, the invaluable contributions of the elderly are honoured, fostering a culture of respect, appreciation, and comprehensive support, ensuring that the senior years are indeed filled with growth, connection, and a profound sense of purpose. ==See also== # [[Motivation and emotion/Book/2014/Ageing and emotion|Ageing and emotion]] (Book chapter, 2014) # [[Motivation and emotion/Book/2023/Ageing and motivation|Ageing and motivation]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Nudge_theory_and_sedentary_behaviour|Nudge theory and sedentary behaviour]] (Book chapter, 2023) # [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) # [[Motivation and emotion/Book/2023/Death_and_meaning_in_life|Death and meaning in life]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Community_resilience|Community resilience]] (Book chapter, 2023) == References == {{Hanging indent|1= Ball, K., Berch, D., Helmers, K., Jobe, J., Leveck, M., Marsiske, M., Morris, J., Rebok, G., Smith, D., Tennstedt, S., Unverzagt, F., & Willis, S. (2002). Effects of cognitive training interventions with older adults. ''JAMA'', ''288''(18), 2271–2281. https://doi.org/10.1001/jama.288.18.2271 Bengtson, V. L. (2001). Beyond the nuclear family: The increasing importance of multigenerational bonds. ''Journal of Marriage and Family'', ''63''(1), 1–16. https://doi.org/10.1111/j.1741-3737.2001.00001.x Bonanno, G. A., Wortman, C. B., & Nesse, R. M. (2004). Prospective patterns of resilience and maladjustment during widowhood. ''Psychology and Aging'', ''19''(2), 260–271. https://doi.org/10.1037/0882-7974.19.2.260 Cacioppo, J. T., Hughes, M. E., Waite, L. J., Hawkley, L. C., & Thisted, R. A. (2006). Loneliness as a specific risk factor for depressive symptoms: Cross-sectional and longitudinal analyses. ''Psychology and Aging'', ''21''(1), 140–151. https://doi.org/10.1037/0882-7974.21.1.140 Carstensen, L. L., Isaacowitz, D. M., & Charles, S. T. (1999). Taking time seriously: A theory of socioemotional selectivity. ''American Psychologist'', ''54''(3), 165–181. https://doi.org/10.1037/0003-066X.54.3.165 Charles, S. T., & Carstensen, L. L. (2010). Social and emotional aging. ''Annual Review of Psychology'', ''61'', 383–409. https://doi.org/10.1146/annurev.psych.093008.100448 Chodzko-Zajko, W. J., Proctor, D. N., Fiatarone Singh, M. A., Minson, C. T., Nigg, C. R., Salem, G. J., & Skinner, J. S. (2009). Exercise and physical activity for older adults. ''Medicine & Science in Sports & Exercise'', ''41''(7), 1510–1530. https://doi.org/10.1249/MSS.0b013e3181a0c95c Cohen, G. D., Perlstein, S., Chapline, J., Kelly, J., Firth, K. M., & Simmens, S. (2006). The impact of professionally conducted cultural programs on the physical health, mental health, and social functioning of older adults. ''The Gerontologist'', ''46''(6), 726–734. https://doi.org/10.1093/geront/46.6.726 Colcombe, S., & Kramer, A. F. (2003). Fitness effects on the cognitive function of older adults: A meta-analytic study. ''Psychological Science'', ''14''(2), 125–130. https://doi.org/10.1111/1467-9280.t01-1-01430 Emmons, R. A., & McCullough, M. E. (2003). Counting blessings versus burdens: An experimental investigation of gratitude and subjective well-being in daily life. ''Journal of Personality and Social Psychology'', ''84''(2), 377–389. https://doi.org/10.1037/0022-3514.84.2.377 Greenfield, E. A., & Marks, N. F. (2004). Formal volunteering as a protective factor for older adults' psychological well-being. ''The Journals of Gerontology Series B: Psychological Sciences and Social Sciences'', ''59''(5), S258–S264. https://doi.org/10.1093/geronb/59.5.S258 Helliwell, J. F., Layard, R., & Sachs, J. (2013). World happiness report. ''United Nations Sustainable Development Solutions Network''. http://eprints.lse.ac.uk/47487/ Hultsch, D. F., Hertzog, C., Small, B. J., & Dixon, R. A. (1999). Use it or lose it: Engaged lifestyle as a buffer of cognitive decline in aging? ''Psychology and Aging'', ''14''(2), 245–263. https://doi.org/10.1037/0882-7974.14.2.245 Keyes, C. L. M. (2007). Promoting and protecting mental health as flourishing: A complementary strategy for improving national mental health. ''American Psychologist'', ''62''(2), 95–108. https://doi.org/10.1037/0003-066X.62.2.95 Liu, C. J., & Latham, N. K. (2009). Progressive resistance strength training for improving physical function in older adults. ''The Cochrane Database of Systematic Reviews'', (3), CD002759. https://doi.org/10.1002/14651858.CD002759.pub2 Martin, R. A., Kuiper, N. A., Olinger, L. J., & Dance, K. A. (1993). Humor, coping with stress, self-concept, and psychological well-being. ''Humor-International Journal of Humor Research'', ''6''(1), 89–104. https://doi.org/10.1515/humr.1993.6.1.89 Park, D. C., & Bischof, G. N. (2013). The aging mind: neuroplasticity in response to cognitive training. ''Dialogues in Clinical Neuroscience'', ''15''(1), 109–119. https://doi.org/10.31887/DCNS.2013.15.1/dpark Park, D. C., Lautenschlager, G., Hedden, T., Davidson, N. S., Smith, A. D., & Smith, P. K. (2002). Models of visuospatial and verbal memory across the adult life span. ''Psychology and Aging'', ''17''(2), 299–320. https://doi.org/10.1037/0882-7974.17.2.299 Paterson, D. H., & Warburton, D. E. (2010). Physical activity and functional limitations in older adults: a systematic review related to Canada's physical activity guidelines. ''International Journal of Behavioral Nutrition and Physical Activity'', ''7''(1), 1–22. https://doi.org/10.1186/1479-5868-7-38 Peterson, C., Ruch, W., Beermann, U., Park, N., & Seligman, M. E. P. (2007). Strengths of character, orientations to happiness, and life satisfaction. ''The Journal of Positive Psychology'', ''2''(3) 149–156. https://doi.org/10.1080/17439760701228938 Reid, K. J., Baron, K. G., Lu, B., Naylor, E., Wolfe, L., & Zee, P. C. (2010). Aerobic exercise improves self-reported sleep and quality of life in older adults with insomnia. ''Sleep Medicine'', ''11''(9), 934–940. https://doi.org/10.1016/j.sleep.2010.04.014 Ryan, R. M., & Deci, E. L. (2018). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. The Guildford Press. Seligman, M. E. P. (2011). ''Flourish''. William Heinnemann. Seligman, M. E. P., Steen, T. A., Park, N., & Peterson, C. (2005). Positive psychology progress: Empirical validation of interventions. ''American Psychologist'', ''60''(5), 410. https://doi.org/10.1037/0003-066X.60.5.410 Sin, N. L., & Lyubomirsky, S. (2009). Enhancing well-being and alleviating depressive symptoms with positive psychology interventions: A practice-friendly meta-analysis. ''Journal of Clinical Psychology'', ''65''(5), 467–487. https://doi.org/10.1002/jclp.20593 Southwick, S. M., & Charney, D. S. (2012) The science of resilience: Implications for the prevention and treatment of depression. ''Science'', ''338''(6103), 79–82. https://doi.org/10.1126/science.1222942 }} == External links == # [https://youtu.be/Is6WvYAM3gg?si=KWHkkgmP47QQDf0O 100-year olds' guide to living your best life] (Allure; YouTube) # [https://www.bluezones.com/2016/11/power-9/ Blue zones power 9: Lifestyle habits of the world’s healthiest, longest-lived people] (bluezones.com) # [https://ppc.sas.upenn.edu/ Positive psychology center] (University of Pennsylvania) # [https://youtu.be/bPBJJ-lxsXA?si=f2yDgG9X6lqiw4Cp The secret to successful aging] (Cathleen Toomey; TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:{{#titleparts:{{PAGENAME}}|3}}/Top]] [[Category:Motivation and emotion/Book/Ageing]] [[Category:Motivation and emotion/Book/Flourishing]] ir2fk3l5k1di552siac7frncjakxany 2803459 2803456 2026-04-08T02:47:03Z Jtneill 10242 /* Overview */ + image for Figure 2 2803459 wikitext text/x-wiki {{title|Flourishing in the elderly:<br>How can psychological flourishing be supported in the elderly?}} {{MECR3|1=https://youtu.be/yUAuPhv5S4o}} __TOC__ ==Overview== <br> {{RoundBoxTop|theme=13}} [[File:Elderly Woman, B&W image by Chalmers Butterfield.jpg|thumb|157x196px|'''Figure 1'''. ''Sarah ...''. (image no longer visible after April 2025 due to Wiki Commons feedback and user request for deletion)]] '''''Can you relate''''' ... Sarah, aged 85, begins her day in a house echoing with memories. Faded photographs on the mantelpiece showcase her once pivotal role in the community. From hosting gatherings to being an active voice at the local council, she had always championed causes dear to her. But now, an overwhelming quietness envelops her heart, and she has lost touch with her sense of purpose. Time has seen her closest friends move or pass away. Family visits, once frequent and filled with laughter, have become increasingly rare. This growing isolation weighs on Sarah, exacerbated by a society that seems to prioritise youth over experience. The fast-paced technological world further alienates her; smart devices and social media platforms feel foreign, exacerbating her deep-seated fear of irrelevance in a world that is rapidly evolving without her. In a society increasingly centred on youth, how can seniors like Sarah reclaim their vitality and sense of meaning? {{RoundBoxBottom}} <br> [[File:Elderly Couple Eating.jpg|thumb|250x250px|'''Figure 2'''. ''An elderly couple ...'' (image no longer visible after March 2026 due to Wiki Commons feedback and user request for deletion)]] The [[wikipedia:Ageing#Successful_ageing|ageing]] population represents an invaluable repository of [[wikipedia:Wisdom|wisdom]], [[wikipedia:Experience|experience]], and [[wikipedia:Insight|insight]] (see Figure 2). Yet, many elderly individuals, much like Sarah, face challenges related to purpose, meaning, and overall psychological [[wikipedia:Well-being|well-being]]. This raises a question; can the elderly attain a state of psychological [[Motivation and emotion/Book/2018/Flourishing|flourishing]], even in the face of age-related adversities? The domain of [[wikipedia:Positive_psychology|positive psychology]] offers evidence-based strategies. Read on to explore what these strategies are and how they can be applied to support seniors in their journey toward a life filled with purpose, meaning, and joy. Specifically, psychological flourishing in the elderly may be supported through a combination of the following four elements, especially when tailored to individual needs and preferences: # [[wikipedia:Social_support|Social engagement]] and meaningful activities such as community involvement or [[wikipedia:Hobby#Psychological_role|hobbies]] (Helliwell et al., 2013) # Mental stimulation via [[wikipedia:Brain_training|cognitive training]] programs can sustain mental acuity and may delay cognitive decline (Ball et al., 2002) # [[Motivation and emotion/Book/2013/Motivating the elderly to exercise|Physical activity]] using exercises tailored to the elderly improve mood and cognitive function (Colcombe & Kramer, 2003) # Incorporating [[Motivation and emotion/Book/2020/Coping and emotion|positive coping strategies]] from positive psychology techniques, such as gratitude exercises, builds [[wikipedia:Psychological_resilience|resilience]] and satisfaction (Seligman et al., 2005) <br> {{RoundBoxTop|theme=3}} [[File:Optics stub.png|right|60px|]] '''Focus questions:''' * What is psychological flourishing? * Why is psychological flourishing important for the elderly? * How does positive psychology foster psychological flourishing? * What can seniors do to flourish? {{RoundBoxBottom}} <br> == What is psychological flourishing? == Psychological flourishing, a term frequently linked to positive psychology, has gained prominence over the years. For elderly individuals, understanding and achieving psychological flourishing is paramount, considering the myriad challenges they confront during this stage of life (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theory, such as self-determination theory and resilience theory, combine with empirical evidence to underpin the suggested four-element model. It is also essential to define and distinguish psychological flourishing from broader notions of general flourishing and well-being. Through understanding this distinction, the unique facets of flourishing and its potential impact on the lives of older individuals become evident. Common misconceptions about psychological flourishing are also addressed, ensuring clarity in subsequent discussions. Psychological flourishing, a concept closely associated with positive psychology, has gained prominence in recent years. For elderly individuals, comprehending and attaining psychological flourishing is crucial, given the diverse challenges encountered in this life stage (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theories such as self-determination theory and resilience theory, as well as empirical evidence, will be used to establish a four-element model of psychological flourishing. Additionally, it is vital to delineate psychological flourishing from broader concepts of general flourishing and well-being. By grasping these distinctions, the unique facets of flourishing and their potential transformative impacts on the lives of older individuals become evident. Common misconceptions about psychological flourishing will also be addressed to ensure clarity in the discussions that follow. === Definitions and distinctions === Psychological flourishing, often simply termed "flourishing", signifies a pinnacle in human functioning. It represents a well-being state that extends beyond merely being free from distress or psychological issues. More than mere survival, flourishing connotes thriving, excelling, and feeling an intense sense of purpose and contentment (Seligman et al., 2005). This condition transcends mere happiness or the absence of mental illness (Sin & Lyubomirsky, 2009). Flourishing provides a holistic view, emphasising positive human functioning across various areas, from relationships to personal growth and purpose. It captures positive emotions, a sense of engagement, strong social connections, and a profound understanding of life's meaning. Distinct from general well-being, flourishing underscores not just feeling good but also functioning effectively, accentuating both [[wikipedia:Hedonic_motivation|hedonic]] (feelings about life) and [[wikipedia:Eudaimonia|eudaimonic]] (functioning in life) well-being facets (Keyes, 2007). Positive psychology’s well-being theory, also known as the PERMA model, conceptualises well-being as a state of comfort, health, or happiness (Seligman, 2011). The model comprises five elements: positive emotion, engagement, positive relationships, meaning, and accomplishment (Seligman, 2011). However, psychological flourishing transcends the ‘feeling good’ aspect of well-being. It encompasses optimal functioning in daily life, incorporating the key principles of self-determination theory: autonomy, relatedness, and competence. === Importance in the context of ageing === Ageing often elicits feelings of apprehension and resignation, influenced by perceptions of decline or limitation, both physically and mentally. Nevertheless, research indicates that many elderly individuals experience periods of growth, insight, and enhanced well-being. For example, the Adult Development and Enrichment Project (ADEPT), conducted by researchers at Pennsylvania State University, underscored the potential for intellectual and emotional growth even in advanced age (Hultsch et al., 1999). Psychological flourishing becomes paramount in this context, assisting the elderly in navigating challenges, capitalising on opportunities, and making meaningful contributions to their communities and personal lives, thereby maintaining their autonomy, relatedness, and competence. Psychological flourishing challenges these notions of decline or limitation, highlighting the profound possibilities for growth, enrichment, and depth in later life (Paterson & Warburton, 2010). Embracing flourishing can significantly transform the way elderly individuals perceive their later years, encouraging them to view this period as an opportunity for renewed purpose, deepened relationships, and the cultivation of new passions or interests. Furthermore, as elderly individuals inevitably face challenges, resilience theory provides a framework for developing a mindset rooted in flourishing (Southwick & Charney, 2012; Bonanno et al., 2004). Such a capacity and mindset act as a solid foundation, empowering seniors to approach obstacles with resilience and poise (Seligman et al., 2005). === Misconceptions === Several misconceptions exist regarding the concept of psychological flourishing. Contrary to common belief, it does not imply a perpetual state of happiness or a life devoid of adversity. Psychological flourishing is not about the absence of negative emotions or challenges. Instead, it focuses on cultivating psychological tools and strategies to navigate adversity, empowering individuals to thrive amidst difficulties (Carstensen et al., 1999). Clarifying these misconceptions paves the way for a deeper understanding of how older individuals can achieve and maintain psychological flourishing. To truly grasp psychological flourishing in older adults, it is crucial to debunk prevalent myths about the ageing process. If left unchallenged, these misconceptions could deter the implementation of strategies promoting well-being in later life (Carstensen et al., 1999; Charles & Carstensen, 2010). Table 1 presents evidence-based counters to these myths, highlighting the potential for growth and vitality among the elderly. {| class="wikitable" |+ Table 1. ''Dispelling Common Misconceptions about the Ageing Process''<br> |- ! Myth !! Fact |- | Ageing leads to inevitable cognitive and emotional decline. || Ageing can offer growth opportunities with the right strategies. |- | Ageing means mental and physical decline. || Many older adults maintain high cognitive and physical activity levels with suitable exercises (Park et al., 2002; Hultsch et al., 1999). |- | Social withdrawal is an ageing inevitability. || While social circles might decrease in size, relationship quality often improves, and staying socially engaged is beneficial (Carstensen et al., 1999; Charles & Carstensen, 2010). |- | Physical activity is risky for older adults. || Moderate physical activity improves mental and physical health in the elderly, refuting the risk myth (Paterson & Warburton, 2010; Chodzko-Zajko et al., 2009). |- | Positive thinking is naive. || Adopting a positive mindset is empirically supported, and positive psychology boosts well-being in the elderly (Seligman et al., 2005; Sin & Lyubomirsky, 2009). |} == Significance of psychological flourishing for the elderly == Psychological flourishing holds a special significance for the elderly (Seligman, 2011). As they navigate the complexities of ageing, fostering a sense of purpose, joy, and well-being is essential (Bonanno et al., 2004). The importance of psychological flourishing for older adults lies in its comprehensive contributions to their emotional, cognitive, and physical domains (Emmons & McCullough, 2003; Hultsch et al., 1999). Drawing upon current research, the following discussion explores the transformative effects of a flourishing mindset. === Emotional benefits === Ageing introduces challenges that can be emotionally strenuous. However, research such as that by Peterson et al. (2007) indicates that character strengths, including love, gratitude, and hope, can significantly enhance emotional well-being. A good sense of humour provides further enhancement to emotional well-being according to research conducted by Martin et al. (1993). Flourishing ensures that the elderly possess the emotional resilience required to confront challenges, nurturing feelings of contentment and fulfilment (Bonanno et al., 2004). A senior who flourishes, for instance, might derive profound joy from meaningful relationships or experience a deep sense of gratitude for life's journey. This emotional equilibrium not only improves overall life quality but also acts as a safeguard against stress, rendering the ageing experience more rewarding (Bengtson, 2001; Carstensen et al., 1999). === Cognitive benefits === Psychological flourishing has discernible effects on cognitive function. A flourishing mind is an active one, continuously engaged in stimulating activities. Various researchers have observed that aspects like mental stimulation and engagement in activities such as walking, cycling, and sports play a role in cognitive maintenance (Bonanno et al., 2004; Emmons & McCullough, 2003; Hultsch et al., 1999). Cross-sectional and retrospective studies, although lacking direct causation evidence, highlight the correlation between physical activity and cognitive function (Hultsch et al., 1999; Paterson & Warburton, 2010). Moreover, character strengths such as curiosity and zest, as pinpointed by Peterson et al. (2007), equip the elderly with a passion for learning, keeping their cognitive faculties sharp and agile. It is a proactive stance against cognitive decline, ensuring that the elderly remain mentally active and engaged. === Impact on physical health and longevity === The connection between the mind and body is profound, particularly in the context of ageing. Research consistently underscores the positive effects of physical activity on functional outcomes, with an emphasis on the benefits of aerobic activities and structured exercise programs (Greenfield & Marks, 2004; Paterson & Warburton, 2010). Notably, regular participation in these activities is linked to decreased risks of functional impairments. Beyond the immediate physical benefits, psychological flourishing, with its focus on positive behaviours and proactive approaches, may also enhance longevity (Bengtson, 2001; Peterson et al. 2007). For example, seniors who cultivate qualities such as gratitude, as identified in studies by Peterson et al. (2007), are likely to adopt more health-promoting behaviours. This symbiotic relationship between a flourishing mind and a healthy body underscores the importance of psychological well-being in the elderly. {{RoundBoxTop|theme=6}} <quiz display=simple> {Imagine you are chatting with a friend about the advantages of psychological flourishing in older adults. Which of the following is NOT a benefit of psychological flourishing in the elderly? |type="()"} - Enhanced emotional resilience to confront challenges + Safeguard against financial issues - Improved overall life quality - Acts as a safeguard against stress {A senior with a flourishing mindset might derive profound joy from meaningful relationships and experience a deep sense of gratitude for life's journey. |type="()"} + True - False {Research indicates that seniors who cultivate qualities like gratitude are more likely to: |type="()"} - Avoid social interactions - Abstain from any form of physical activity + Adopt additional health-promoting behaviours - Experience rapid cognitive decline </quiz> {{RoundBoxBottom}} == The role of positive psychology == Positive psychology represents a rapidly growing field, concentrating on the cultivation of individual strengths, virtues, and peak human functioning, as highlighted by Seligman (2011). For elderly individuals, this approach provides a comprehensive viewpoint, tackling the intrinsic challenges associated with ageing while actively encouraging a flourishing life that extends past mere existence. Southwick and Charney's (2012) influential research underscores resilience, an essential aspect of positive psychology's resilience theory, as crucial in managing stress and trauma. This section explores relevant core principles of positive psychology that, illustrate its substantial utility in supporting psychological flourishing among seniors. === Foundational principles and theories === Positive psychology fundamentally examines the conditions and processes essential for optimal functioning, as investigated by Southwick and Charney (2012). It transforms the traditional focus on deficits and disorders, advocating instead for the amplification of strengths, positive emotions, and resilience. These elements are crucial for seniors, offering invaluable guidance through the complexities of later life. Core theoretical frameworks within this field, including resilience theory, self-determination theory, and well-being theory, shed light on various avenues available for seniors to discover meaning, purpose, and an elevated sense of well-being, even amidst the challenges life presents. * '''Resilience theory''' in psychology delves into the complex processes that enable individuals to adapt positively in the face of adversity. Recognising resilience as a dynamic and developmental capacity, it encompasses both internal cognitive-affective mechanisms, such as emotion regulation and a robust sense of self-efficacy, and external factors like strong social support networks. This framework underscores the notion that resilience is not an innate trait; rather, it is a skill subject to cultivation. The theory also integrates the concept of post-traumatic growth, shedding light on how challenging experiences can contribute to personal development and a deeper sense of meaning in life. By considering individual differences and the developmental context, resilience theory provides a nuanced understanding of adaptive functioning, guiding interventions that aim to enhance resilience and promote positive psychological outcomes (Southwick & Charney, 2012; Bonanno et al., 2004). * '''Self-determination theory''' (SDT) stands as a prominent psychological framework elucidating the intrinsic and extrinsic factors driving human motivation and behaviour. Developed by Deci and Ryan, the theory posits that the fulfillment of three core psychological needs—autonomy, competence, and relatedness—is paramount for optimal functioning and well-being (Ryan & Deci, 2018). Autonomy pertains to the sense of volition and self-governance in one’s actions, competence encompasses the need to master tasks and learn, and relatedness refers to the desire for meaningful connections and belongingness. In the context of flourishing, particularly among seniors, SDT offers valuable insights into how supportive environments and interventions can be crafted to nurture these fundamental needs. By emphasising the integral role of internal motivation and the social context in psychological well-being, SDT provides a nuanced and comprehensive understanding of the pathways leading to enhanced flourishing and life satisfaction. This theory, therefore, serves as a pivotal guide in the quest to foster resilient, connected, and self-determined lives in the elderly population. * '''Well-being theory''', conceptualised by Martin Seligman, represents a pivotal shift in psychological thought, extending the focus from the mere alleviation of distress to the active cultivation of optimal living. Seligman (2011) introduced the PERMA model, which encapsulates five integral components of well-being: positive emotion, engagement, positive relationships, meaning, and accomplishment. Positive emotion emphasises the value of fostering joy and contentment, while engagement stresses the importance of deep absorption in activities, invoking a state of flow. Positive relationships highlight the necessity of nurturing supportive and enriching connections. Meaning pertains to pursuing a purpose larger than oneself, and accomplishment encompasses striving for mastery and achievement. For the elderly, these dimensions provide a comprehensive blueprint for flourishing, guiding interventions and practices aimed at enhancing life quality and fulfilment. Well-being theory thus stands as a foundational pillar in positive psychology, offering a nuanced and holistic understanding of the factors that contribute to a life well-lived, irrespective of one’s age. === Psychological functioning and its role === Research conducted by Southwick and Charney (2012) underscores the integral role of psychological resilience in combatting depression—a condition prevalent among the elderly (16.1% of 65+-year-old Australians in [https://www.abs.gov.au/statistics/health/health-conditions-and-risks/health-conditions-prevalence/latest-release 2020–21]). Resilience does not merely denote bouncing back from adversity but signifies thriving amidst it. The elderly, with a lifetime of experiences, possess unique strengths and coping mechanisms. By harnessing these, and understanding neurobiological and psychosocial factors, there is an avenue for improved psychological functioning, ultimately contributing to enhanced flourishing. === The resilience connection === Several studies highlight that resilience is intricately linked to genetics, environment, neurobiology, and psychosocial factors (Helliwell & Sachs, 2013; Seligman, 2011; Southwick & Charney, 2012). For seniors, building resilience becomes a pivotal tool in managing age-related stressors and traumas. Embracing resilience does not mean avoiding challenges but rather developing a robust toolkit to face them head-on. With strategies encompassing positive emotions, social support, coping skills, and more, resilience offers a protective layer, allowing seniors to remain buoyant in the face of life's trials. === Proactive approaches and interventions === Promoting flourishing among the elderly necessitates actionable steps grounded in empirical evidence. Southwick and Charney (2012) advocate for interventions such as modifying the biological and psychosocial environment, strengthening social support networks, enhancing cognitive engagements, and boosting physical health through aerobic exercises. It is crucial to tailor these approaches to the distinct needs of seniors, taking into account their physiological, cognitive, and socio-emotional states. When informed by positive psychology, the amalgamation of these interventions paves the way for a life marked by heightened contentment, purpose, and joy in the latter years. Figure 3 illustrates how the integration of these supportive interventions correlates with increased levels of psychological flourishing among the elderly.<br> [[File:Psychological Flourishing - amalgamation model.png|center|thumb|400x400px|'''Figure 3'''. ''The cumulative impact of multiple interventions for supporting flourishing in the elderly''.]] == What can seniors do to flourish? == Psychological flourishing in seniors is essential for their well-being and quality of life. This section delves into practical, evidence-based strategies derived from positive psychology to foster flourishing among the elderly. It emphasises the importance of social engagement, mental stimulation, physical activity, and positive coping techniques. Tailored to individual needs, these approaches aim to enhance purpose, joy, and meaningful connections, addressing the unique challenges faced by seniors in their pursuit of a fulfilling life. === Social engagement and meaningful activities === * Building social connections rejuvenates the spirit and enhances well-being. * Social relationships have been found to significantly influence mental and emotional well-being in the elderly. A strong social network can reduce feelings of loneliness and depression, thereby promoting psychological flourishing (Cacioppo et al., 2006). * Forming bonds across different age groups can be mutually beneficial and specifically aid the elderly in feeling more connected and less isolated (Bengtson, 2001). * Engaging in community activities or volunteering (see Figure 4) has been shown to provide a sense of purpose and improve mental health outcomes for seniors (Greenfield & Marks, 2004). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': When was the last time you made a new friend? What activities can help you meet new people? :''Suggestions'': Engage in community activities, reconnect with old friends, or even consider pet ownership.<br> ::Establish regular family visits or calls, attend local gatherings, or join clubs focused on specific interests. |} [[File:Deeply engrossed in puzzle.png|thumb|317x317px|'''Figure 5'''''.'' ''An elderly woman deeply engrossed in her daily crossword puzzle; an excellent form of mental stimulation''.]] === Mental stimulation === * Continuous learning has been linked to cognitive vitality and emotional well-being (Park & Bischof, 2013). * Activities that require creativity, such as painting or music, not only stimulate the brain but also contribute to a greater sense of purpose and joy, enhancing the quality of life (Cohen et al., 2006). * Research indicates that lifelong learning and mental stimulation can help prevent cognitive decline and improve overall psychological well-being. Older adults who engage in mentally stimulating activities (see Figure 5) report higher levels of happiness and lower levels of depression (Hultsch et al., 1999). * More recent research underscores the notion that the ageing brain is capable of new neural connections when subjected to novel learning experiences, enhancing cognitive and emotional well-being (Park & Bischof, 2013). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Take up a new hobby, join an interesting class, or simply read a new book.<br> ::Explore online courses tailored for seniors or consider group-based activities to foster intellectual engagement. |} === Physical activity === [[File:TaiChi-group.png|thumb|300x280px|'''Figure 6'''''.'' ''A group of elderly enjoying a tai chi class tailored for maintaining mobility and easing arthritis symptoms''.]] * Regular physical activity boosts mental health and protects against age-related ailments such as heart disease, osteoporosis, and [https://en.wikipedia.org/wiki/Sarcopenia sarcopenia] (loss of muscle mass). * Several studies show that regular physical exercise can improve cognitive function, thereby supporting not only physical but also mental well-being (Colcombe & Kramer, 2003). * Group exercise activities like tai chi (see Figure 6) or water aerobics offer not only physical benefits but also social interaction, which can further contribute to psychological well-being (Liu & Latham, 2009). * Improved sleep through regular physical activity is correlated with better mood and mental health, providing another pathway to psychological flourishing (Reid et al., 2010). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Incorporate simple exercises into your daily routine, such as ::walking, yoga, or dancing, or consider joining a senior-friendly exercise group.<br> :''Please note:'' Always consult your healthcare professional before starting any new exercise regimen. |} [[File:Meditating man-solo.png|thumb|317x317px|'''Figure 7'''''.'' ''An elderly man practicing guided meditation; an excellent positive coping strategy''.]] === Positive coping strategies === * Develop resilience against challenges by adopting positive coping mechanisms such as meditation (see Figure 7), relaxation techniques, and engaging in spirituality. * Positive psychology interventions focus on strengths and virtues and have shown efficacy in improving well-being and reducing depressive symptoms in older adults (Seligman et al., 2005). * Incorporating gratitude into daily routines has been associated with positive emotional states, greater well-being, and better physical health in older adults (Emmons & McCullough, 2003). * Utilising humor is shown to not only uplift mood but also serve as an effective coping strategy for stress and life challenges. This is particularly relevant for elderly individuals who may face various forms of age-related adversity (Martin et al., 1993). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a past challenge. How did you cope? How can you refine this strategy? :''Suggestions'': Embrace activities that foster mindfulness and reflection, such as ::meditation, gratitude journaling, or joining a support group.<br> |} {{RoundBoxTop|theme=13}} [[File:Elderly man-zest for life-mentoring.png|thumb|327x182px|'''Figure 8'''. ''John rediscovered his zest for life through mentoring teenagers, a program organised by his local APEX club''.]] '''''Can you relate''''' ... John, at 80, often sat in his armchair, lost in memories of youthful adventures. He believed his golden years had long passed, with each day echoing the sentiments of a vibrant past. Collecting his mail one day, a flyer for the local book club caught his attention, hinting at a promise of engaging discussions. Deciding to join, John discovered that the club was more than just about books; it was a community bridging generational gaps through shared stories. A young woman from the group, impressed by John's vast life experiences, introduced him to a mentoring program hosted by the local [[w:Apex Clubs of Australia|APEX]] club. Through mentoring, John shared his life lessons, offering wisdom and guidance to the younger generation. This exchange rekindled his understanding of the value of his own journey. Far from feeling that his best years were behind him, the interactions brought about a renewed sense of purpose (see Figure 8). Through the book club and mentoring, John not only found a revived passion for literature but also tapped into a deeper zest for life, seeing his age not as a limitation but as a testament to a life rich with experiences. {{RoundBoxBottom}} ==Conclusion== Ageing, despite its inherent challenges, can still be a period rich in growth, connection, and profound meaning. By integrating the principles of positive psychology with tailored interventions, an environment conducive to psychological flourishing in the elderly can be cultivated. Foundational theories such as resilience theory, self-determination theory, and well-being theory offer a robust framework for understanding and nurturing the various facets of flourishing. Translating these theories into practical strategies and interventions empowers seniors to gracefully navigate the complexities of ageing, maintaining resilience and a sustained sense of well-being. Supporting flourishing in the elderly necessitates a commitment to social engagement, enhancing emotional well-being; continuous mental stimulation, promoting cognitive vitality; regular physical activity, supporting both physical and mental health; and the adoption of positive coping strategies to foster resilience. These strategies not only enhance their quality of life but also contribute positively to the broader community, as the elderly impart their wisdom, experience, and emotional stability. Furthermore, dispelling misconceptions surrounding ageing and psychological flourishing is paramount. Challenging societal stereotypes and adopting a holistic view of ageing is essential, acknowledging the potential for growth, development, and fulfilment in this life stage. In conclusion, supporting psychological flourishing in the elderly is a multifaceted endeavour. It demands a collaborative effort from individuals, communities, and society at large, aiming to create environments that nurture the emotional, cognitive, and physical aspects of well-being. In doing so, the invaluable contributions of the elderly are honoured, fostering a culture of respect, appreciation, and comprehensive support, ensuring that the senior years are indeed filled with growth, connection, and a profound sense of purpose. ==See also== # [[Motivation and emotion/Book/2014/Ageing and emotion|Ageing and emotion]] (Book chapter, 2014) # [[Motivation and emotion/Book/2023/Ageing and motivation|Ageing and motivation]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Nudge_theory_and_sedentary_behaviour|Nudge theory and sedentary behaviour]] (Book chapter, 2023) # [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) # [[Motivation and emotion/Book/2023/Death_and_meaning_in_life|Death and meaning in life]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Community_resilience|Community resilience]] (Book chapter, 2023) == References == {{Hanging indent|1= Ball, K., Berch, D., Helmers, K., Jobe, J., Leveck, M., Marsiske, M., Morris, J., Rebok, G., Smith, D., Tennstedt, S., Unverzagt, F., & Willis, S. (2002). Effects of cognitive training interventions with older adults. ''JAMA'', ''288''(18), 2271–2281. https://doi.org/10.1001/jama.288.18.2271 Bengtson, V. L. (2001). Beyond the nuclear family: The increasing importance of multigenerational bonds. ''Journal of Marriage and Family'', ''63''(1), 1–16. https://doi.org/10.1111/j.1741-3737.2001.00001.x Bonanno, G. A., Wortman, C. B., & Nesse, R. M. (2004). Prospective patterns of resilience and maladjustment during widowhood. ''Psychology and Aging'', ''19''(2), 260–271. https://doi.org/10.1037/0882-7974.19.2.260 Cacioppo, J. T., Hughes, M. E., Waite, L. J., Hawkley, L. C., & Thisted, R. A. (2006). Loneliness as a specific risk factor for depressive symptoms: Cross-sectional and longitudinal analyses. ''Psychology and Aging'', ''21''(1), 140–151. https://doi.org/10.1037/0882-7974.21.1.140 Carstensen, L. L., Isaacowitz, D. M., & Charles, S. T. (1999). Taking time seriously: A theory of socioemotional selectivity. ''American Psychologist'', ''54''(3), 165–181. https://doi.org/10.1037/0003-066X.54.3.165 Charles, S. T., & Carstensen, L. L. (2010). Social and emotional aging. ''Annual Review of Psychology'', ''61'', 383–409. https://doi.org/10.1146/annurev.psych.093008.100448 Chodzko-Zajko, W. J., Proctor, D. N., Fiatarone Singh, M. A., Minson, C. T., Nigg, C. R., Salem, G. J., & Skinner, J. S. (2009). Exercise and physical activity for older adults. ''Medicine & Science in Sports & Exercise'', ''41''(7), 1510–1530. https://doi.org/10.1249/MSS.0b013e3181a0c95c Cohen, G. D., Perlstein, S., Chapline, J., Kelly, J., Firth, K. M., & Simmens, S. (2006). The impact of professionally conducted cultural programs on the physical health, mental health, and social functioning of older adults. ''The Gerontologist'', ''46''(6), 726–734. https://doi.org/10.1093/geront/46.6.726 Colcombe, S., & Kramer, A. F. (2003). Fitness effects on the cognitive function of older adults: A meta-analytic study. ''Psychological Science'', ''14''(2), 125–130. https://doi.org/10.1111/1467-9280.t01-1-01430 Emmons, R. A., & McCullough, M. E. (2003). Counting blessings versus burdens: An experimental investigation of gratitude and subjective well-being in daily life. ''Journal of Personality and Social Psychology'', ''84''(2), 377–389. https://doi.org/10.1037/0022-3514.84.2.377 Greenfield, E. A., & Marks, N. F. (2004). Formal volunteering as a protective factor for older adults' psychological well-being. ''The Journals of Gerontology Series B: Psychological Sciences and Social Sciences'', ''59''(5), S258–S264. https://doi.org/10.1093/geronb/59.5.S258 Helliwell, J. F., Layard, R., & Sachs, J. (2013). World happiness report. ''United Nations Sustainable Development Solutions Network''. http://eprints.lse.ac.uk/47487/ Hultsch, D. F., Hertzog, C., Small, B. J., & Dixon, R. A. (1999). Use it or lose it: Engaged lifestyle as a buffer of cognitive decline in aging? ''Psychology and Aging'', ''14''(2), 245–263. https://doi.org/10.1037/0882-7974.14.2.245 Keyes, C. L. M. (2007). Promoting and protecting mental health as flourishing: A complementary strategy for improving national mental health. ''American Psychologist'', ''62''(2), 95–108. https://doi.org/10.1037/0003-066X.62.2.95 Liu, C. J., & Latham, N. K. (2009). Progressive resistance strength training for improving physical function in older adults. ''The Cochrane Database of Systematic Reviews'', (3), CD002759. https://doi.org/10.1002/14651858.CD002759.pub2 Martin, R. A., Kuiper, N. A., Olinger, L. J., & Dance, K. A. (1993). Humor, coping with stress, self-concept, and psychological well-being. ''Humor-International Journal of Humor Research'', ''6''(1), 89–104. https://doi.org/10.1515/humr.1993.6.1.89 Park, D. C., & Bischof, G. N. (2013). The aging mind: neuroplasticity in response to cognitive training. ''Dialogues in Clinical Neuroscience'', ''15''(1), 109–119. https://doi.org/10.31887/DCNS.2013.15.1/dpark Park, D. C., Lautenschlager, G., Hedden, T., Davidson, N. S., Smith, A. D., & Smith, P. K. (2002). Models of visuospatial and verbal memory across the adult life span. ''Psychology and Aging'', ''17''(2), 299–320. https://doi.org/10.1037/0882-7974.17.2.299 Paterson, D. H., & Warburton, D. E. (2010). Physical activity and functional limitations in older adults: a systematic review related to Canada's physical activity guidelines. ''International Journal of Behavioral Nutrition and Physical Activity'', ''7''(1), 1–22. https://doi.org/10.1186/1479-5868-7-38 Peterson, C., Ruch, W., Beermann, U., Park, N., & Seligman, M. E. P. (2007). Strengths of character, orientations to happiness, and life satisfaction. ''The Journal of Positive Psychology'', ''2''(3) 149–156. https://doi.org/10.1080/17439760701228938 Reid, K. J., Baron, K. G., Lu, B., Naylor, E., Wolfe, L., & Zee, P. C. (2010). Aerobic exercise improves self-reported sleep and quality of life in older adults with insomnia. ''Sleep Medicine'', ''11''(9), 934–940. https://doi.org/10.1016/j.sleep.2010.04.014 Ryan, R. M., & Deci, E. L. (2018). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. The Guildford Press. Seligman, M. E. P. (2011). ''Flourish''. William Heinnemann. Seligman, M. E. P., Steen, T. A., Park, N., & Peterson, C. (2005). Positive psychology progress: Empirical validation of interventions. ''American Psychologist'', ''60''(5), 410. https://doi.org/10.1037/0003-066X.60.5.410 Sin, N. L., & Lyubomirsky, S. (2009). Enhancing well-being and alleviating depressive symptoms with positive psychology interventions: A practice-friendly meta-analysis. ''Journal of Clinical Psychology'', ''65''(5), 467–487. https://doi.org/10.1002/jclp.20593 Southwick, S. M., & Charney, D. S. (2012) The science of resilience: Implications for the prevention and treatment of depression. ''Science'', ''338''(6103), 79–82. https://doi.org/10.1126/science.1222942 }} == External links == # [https://youtu.be/Is6WvYAM3gg?si=KWHkkgmP47QQDf0O 100-year olds' guide to living your best life] (Allure; YouTube) # [https://www.bluezones.com/2016/11/power-9/ Blue zones power 9: Lifestyle habits of the world’s healthiest, longest-lived people] (bluezones.com) # [https://ppc.sas.upenn.edu/ Positive psychology center] (University of Pennsylvania) # [https://youtu.be/bPBJJ-lxsXA?si=f2yDgG9X6lqiw4Cp The secret to successful aging] (Cathleen Toomey; TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:{{#titleparts:{{PAGENAME}}|3}}/Top]] [[Category:Motivation and emotion/Book/Ageing]] [[Category:Motivation and emotion/Book/Flourishing]] 5g75tqfmldseigeozt8hy655axz9mtq 2803460 2803459 2026-04-08T02:47:43Z Jtneill 10242 /* Overview */ Remove focus questions image 2803460 wikitext text/x-wiki {{title|Flourishing in the elderly:<br>How can psychological flourishing be supported in the elderly?}} {{MECR3|1=https://youtu.be/yUAuPhv5S4o}} __TOC__ ==Overview== <br> {{RoundBoxTop|theme=13}} [[File:Elderly Woman, B&W image by Chalmers Butterfield.jpg|thumb|157x196px|'''Figure 1'''. ''Sarah ...''. (image no longer visible after April 2025 due to Wiki Commons feedback and user request for deletion)]] '''''Can you relate''''' ... Sarah, aged 85, begins her day in a house echoing with memories. Faded photographs on the mantelpiece showcase her once pivotal role in the community. From hosting gatherings to being an active voice at the local council, she had always championed causes dear to her. But now, an overwhelming quietness envelops her heart, and she has lost touch with her sense of purpose. Time has seen her closest friends move or pass away. Family visits, once frequent and filled with laughter, have become increasingly rare. This growing isolation weighs on Sarah, exacerbated by a society that seems to prioritise youth over experience. The fast-paced technological world further alienates her; smart devices and social media platforms feel foreign, exacerbating her deep-seated fear of irrelevance in a world that is rapidly evolving without her. In a society increasingly centred on youth, how can seniors like Sarah reclaim their vitality and sense of meaning? {{RoundBoxBottom}} <br> [[File:Elderly Couple Eating.jpg|thumb|250x250px|'''Figure 2'''. ''An elderly couple ...'' (image no longer visible after March 2026 due to Wiki Commons feedback and user request for deletion)]] The [[wikipedia:Ageing#Successful_ageing|ageing]] population represents an invaluable repository of [[wikipedia:Wisdom|wisdom]], [[wikipedia:Experience|experience]], and [[wikipedia:Insight|insight]] (see Figure 2). Yet, many elderly individuals, much like Sarah, face challenges related to purpose, meaning, and overall psychological [[wikipedia:Well-being|well-being]]. This raises a question; can the elderly attain a state of psychological [[Motivation and emotion/Book/2018/Flourishing|flourishing]], even in the face of age-related adversities? The domain of [[wikipedia:Positive_psychology|positive psychology]] offers evidence-based strategies. Read on to explore what these strategies are and how they can be applied to support seniors in their journey toward a life filled with purpose, meaning, and joy. Specifically, psychological flourishing in the elderly may be supported through a combination of the following four elements, especially when tailored to individual needs and preferences: # [[wikipedia:Social_support|Social engagement]] and meaningful activities such as community involvement or [[wikipedia:Hobby#Psychological_role|hobbies]] (Helliwell et al., 2013) # Mental stimulation via [[wikipedia:Brain_training|cognitive training]] programs can sustain mental acuity and may delay cognitive decline (Ball et al., 2002) # [[Motivation and emotion/Book/2013/Motivating the elderly to exercise|Physical activity]] using exercises tailored to the elderly improve mood and cognitive function (Colcombe & Kramer, 2003) # Incorporating [[Motivation and emotion/Book/2020/Coping and emotion|positive coping strategies]] from positive psychology techniques, such as gratitude exercises, builds [[wikipedia:Psychological_resilience|resilience]] and satisfaction (Seligman et al., 2005) <br> {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is psychological flourishing? * Why is psychological flourishing important for the elderly? * How does positive psychology foster psychological flourishing? * What can seniors do to flourish? {{RoundBoxBottom}} <br> == What is psychological flourishing? == Psychological flourishing, a term frequently linked to positive psychology, has gained prominence over the years. For elderly individuals, understanding and achieving psychological flourishing is paramount, considering the myriad challenges they confront during this stage of life (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theory, such as self-determination theory and resilience theory, combine with empirical evidence to underpin the suggested four-element model. It is also essential to define and distinguish psychological flourishing from broader notions of general flourishing and well-being. Through understanding this distinction, the unique facets of flourishing and its potential impact on the lives of older individuals become evident. Common misconceptions about psychological flourishing are also addressed, ensuring clarity in subsequent discussions. Psychological flourishing, a concept closely associated with positive psychology, has gained prominence in recent years. For elderly individuals, comprehending and attaining psychological flourishing is crucial, given the diverse challenges encountered in this life stage (Park et al., 2002). The concept of psychological flourishing and its significance, particularly in the context of ageing, will be explored in depth. Psychological theories such as self-determination theory and resilience theory, as well as empirical evidence, will be used to establish a four-element model of psychological flourishing. Additionally, it is vital to delineate psychological flourishing from broader concepts of general flourishing and well-being. By grasping these distinctions, the unique facets of flourishing and their potential transformative impacts on the lives of older individuals become evident. Common misconceptions about psychological flourishing will also be addressed to ensure clarity in the discussions that follow. === Definitions and distinctions === Psychological flourishing, often simply termed "flourishing", signifies a pinnacle in human functioning. It represents a well-being state that extends beyond merely being free from distress or psychological issues. More than mere survival, flourishing connotes thriving, excelling, and feeling an intense sense of purpose and contentment (Seligman et al., 2005). This condition transcends mere happiness or the absence of mental illness (Sin & Lyubomirsky, 2009). Flourishing provides a holistic view, emphasising positive human functioning across various areas, from relationships to personal growth and purpose. It captures positive emotions, a sense of engagement, strong social connections, and a profound understanding of life's meaning. Distinct from general well-being, flourishing underscores not just feeling good but also functioning effectively, accentuating both [[wikipedia:Hedonic_motivation|hedonic]] (feelings about life) and [[wikipedia:Eudaimonia|eudaimonic]] (functioning in life) well-being facets (Keyes, 2007). Positive psychology’s well-being theory, also known as the PERMA model, conceptualises well-being as a state of comfort, health, or happiness (Seligman, 2011). The model comprises five elements: positive emotion, engagement, positive relationships, meaning, and accomplishment (Seligman, 2011). However, psychological flourishing transcends the ‘feeling good’ aspect of well-being. It encompasses optimal functioning in daily life, incorporating the key principles of self-determination theory: autonomy, relatedness, and competence. === Importance in the context of ageing === Ageing often elicits feelings of apprehension and resignation, influenced by perceptions of decline or limitation, both physically and mentally. Nevertheless, research indicates that many elderly individuals experience periods of growth, insight, and enhanced well-being. For example, the Adult Development and Enrichment Project (ADEPT), conducted by researchers at Pennsylvania State University, underscored the potential for intellectual and emotional growth even in advanced age (Hultsch et al., 1999). Psychological flourishing becomes paramount in this context, assisting the elderly in navigating challenges, capitalising on opportunities, and making meaningful contributions to their communities and personal lives, thereby maintaining their autonomy, relatedness, and competence. Psychological flourishing challenges these notions of decline or limitation, highlighting the profound possibilities for growth, enrichment, and depth in later life (Paterson & Warburton, 2010). Embracing flourishing can significantly transform the way elderly individuals perceive their later years, encouraging them to view this period as an opportunity for renewed purpose, deepened relationships, and the cultivation of new passions or interests. Furthermore, as elderly individuals inevitably face challenges, resilience theory provides a framework for developing a mindset rooted in flourishing (Southwick & Charney, 2012; Bonanno et al., 2004). Such a capacity and mindset act as a solid foundation, empowering seniors to approach obstacles with resilience and poise (Seligman et al., 2005). === Misconceptions === Several misconceptions exist regarding the concept of psychological flourishing. Contrary to common belief, it does not imply a perpetual state of happiness or a life devoid of adversity. Psychological flourishing is not about the absence of negative emotions or challenges. Instead, it focuses on cultivating psychological tools and strategies to navigate adversity, empowering individuals to thrive amidst difficulties (Carstensen et al., 1999). Clarifying these misconceptions paves the way for a deeper understanding of how older individuals can achieve and maintain psychological flourishing. To truly grasp psychological flourishing in older adults, it is crucial to debunk prevalent myths about the ageing process. If left unchallenged, these misconceptions could deter the implementation of strategies promoting well-being in later life (Carstensen et al., 1999; Charles & Carstensen, 2010). Table 1 presents evidence-based counters to these myths, highlighting the potential for growth and vitality among the elderly. {| class="wikitable" |+ Table 1. ''Dispelling Common Misconceptions about the Ageing Process''<br> |- ! Myth !! Fact |- | Ageing leads to inevitable cognitive and emotional decline. || Ageing can offer growth opportunities with the right strategies. |- | Ageing means mental and physical decline. || Many older adults maintain high cognitive and physical activity levels with suitable exercises (Park et al., 2002; Hultsch et al., 1999). |- | Social withdrawal is an ageing inevitability. || While social circles might decrease in size, relationship quality often improves, and staying socially engaged is beneficial (Carstensen et al., 1999; Charles & Carstensen, 2010). |- | Physical activity is risky for older adults. || Moderate physical activity improves mental and physical health in the elderly, refuting the risk myth (Paterson & Warburton, 2010; Chodzko-Zajko et al., 2009). |- | Positive thinking is naive. || Adopting a positive mindset is empirically supported, and positive psychology boosts well-being in the elderly (Seligman et al., 2005; Sin & Lyubomirsky, 2009). |} == Significance of psychological flourishing for the elderly == Psychological flourishing holds a special significance for the elderly (Seligman, 2011). As they navigate the complexities of ageing, fostering a sense of purpose, joy, and well-being is essential (Bonanno et al., 2004). The importance of psychological flourishing for older adults lies in its comprehensive contributions to their emotional, cognitive, and physical domains (Emmons & McCullough, 2003; Hultsch et al., 1999). Drawing upon current research, the following discussion explores the transformative effects of a flourishing mindset. === Emotional benefits === Ageing introduces challenges that can be emotionally strenuous. However, research such as that by Peterson et al. (2007) indicates that character strengths, including love, gratitude, and hope, can significantly enhance emotional well-being. A good sense of humour provides further enhancement to emotional well-being according to research conducted by Martin et al. (1993). Flourishing ensures that the elderly possess the emotional resilience required to confront challenges, nurturing feelings of contentment and fulfilment (Bonanno et al., 2004). A senior who flourishes, for instance, might derive profound joy from meaningful relationships or experience a deep sense of gratitude for life's journey. This emotional equilibrium not only improves overall life quality but also acts as a safeguard against stress, rendering the ageing experience more rewarding (Bengtson, 2001; Carstensen et al., 1999). === Cognitive benefits === Psychological flourishing has discernible effects on cognitive function. A flourishing mind is an active one, continuously engaged in stimulating activities. Various researchers have observed that aspects like mental stimulation and engagement in activities such as walking, cycling, and sports play a role in cognitive maintenance (Bonanno et al., 2004; Emmons & McCullough, 2003; Hultsch et al., 1999). Cross-sectional and retrospective studies, although lacking direct causation evidence, highlight the correlation between physical activity and cognitive function (Hultsch et al., 1999; Paterson & Warburton, 2010). Moreover, character strengths such as curiosity and zest, as pinpointed by Peterson et al. (2007), equip the elderly with a passion for learning, keeping their cognitive faculties sharp and agile. It is a proactive stance against cognitive decline, ensuring that the elderly remain mentally active and engaged. === Impact on physical health and longevity === The connection between the mind and body is profound, particularly in the context of ageing. Research consistently underscores the positive effects of physical activity on functional outcomes, with an emphasis on the benefits of aerobic activities and structured exercise programs (Greenfield & Marks, 2004; Paterson & Warburton, 2010). Notably, regular participation in these activities is linked to decreased risks of functional impairments. Beyond the immediate physical benefits, psychological flourishing, with its focus on positive behaviours and proactive approaches, may also enhance longevity (Bengtson, 2001; Peterson et al. 2007). For example, seniors who cultivate qualities such as gratitude, as identified in studies by Peterson et al. (2007), are likely to adopt more health-promoting behaviours. This symbiotic relationship between a flourishing mind and a healthy body underscores the importance of psychological well-being in the elderly. {{RoundBoxTop|theme=6}} <quiz display=simple> {Imagine you are chatting with a friend about the advantages of psychological flourishing in older adults. Which of the following is NOT a benefit of psychological flourishing in the elderly? |type="()"} - Enhanced emotional resilience to confront challenges + Safeguard against financial issues - Improved overall life quality - Acts as a safeguard against stress {A senior with a flourishing mindset might derive profound joy from meaningful relationships and experience a deep sense of gratitude for life's journey. |type="()"} + True - False {Research indicates that seniors who cultivate qualities like gratitude are more likely to: |type="()"} - Avoid social interactions - Abstain from any form of physical activity + Adopt additional health-promoting behaviours - Experience rapid cognitive decline </quiz> {{RoundBoxBottom}} == The role of positive psychology == Positive psychology represents a rapidly growing field, concentrating on the cultivation of individual strengths, virtues, and peak human functioning, as highlighted by Seligman (2011). For elderly individuals, this approach provides a comprehensive viewpoint, tackling the intrinsic challenges associated with ageing while actively encouraging a flourishing life that extends past mere existence. Southwick and Charney's (2012) influential research underscores resilience, an essential aspect of positive psychology's resilience theory, as crucial in managing stress and trauma. This section explores relevant core principles of positive psychology that, illustrate its substantial utility in supporting psychological flourishing among seniors. === Foundational principles and theories === Positive psychology fundamentally examines the conditions and processes essential for optimal functioning, as investigated by Southwick and Charney (2012). It transforms the traditional focus on deficits and disorders, advocating instead for the amplification of strengths, positive emotions, and resilience. These elements are crucial for seniors, offering invaluable guidance through the complexities of later life. Core theoretical frameworks within this field, including resilience theory, self-determination theory, and well-being theory, shed light on various avenues available for seniors to discover meaning, purpose, and an elevated sense of well-being, even amidst the challenges life presents. * '''Resilience theory''' in psychology delves into the complex processes that enable individuals to adapt positively in the face of adversity. Recognising resilience as a dynamic and developmental capacity, it encompasses both internal cognitive-affective mechanisms, such as emotion regulation and a robust sense of self-efficacy, and external factors like strong social support networks. This framework underscores the notion that resilience is not an innate trait; rather, it is a skill subject to cultivation. The theory also integrates the concept of post-traumatic growth, shedding light on how challenging experiences can contribute to personal development and a deeper sense of meaning in life. By considering individual differences and the developmental context, resilience theory provides a nuanced understanding of adaptive functioning, guiding interventions that aim to enhance resilience and promote positive psychological outcomes (Southwick & Charney, 2012; Bonanno et al., 2004). * '''Self-determination theory''' (SDT) stands as a prominent psychological framework elucidating the intrinsic and extrinsic factors driving human motivation and behaviour. Developed by Deci and Ryan, the theory posits that the fulfillment of three core psychological needs—autonomy, competence, and relatedness—is paramount for optimal functioning and well-being (Ryan & Deci, 2018). Autonomy pertains to the sense of volition and self-governance in one’s actions, competence encompasses the need to master tasks and learn, and relatedness refers to the desire for meaningful connections and belongingness. In the context of flourishing, particularly among seniors, SDT offers valuable insights into how supportive environments and interventions can be crafted to nurture these fundamental needs. By emphasising the integral role of internal motivation and the social context in psychological well-being, SDT provides a nuanced and comprehensive understanding of the pathways leading to enhanced flourishing and life satisfaction. This theory, therefore, serves as a pivotal guide in the quest to foster resilient, connected, and self-determined lives in the elderly population. * '''Well-being theory''', conceptualised by Martin Seligman, represents a pivotal shift in psychological thought, extending the focus from the mere alleviation of distress to the active cultivation of optimal living. Seligman (2011) introduced the PERMA model, which encapsulates five integral components of well-being: positive emotion, engagement, positive relationships, meaning, and accomplishment. Positive emotion emphasises the value of fostering joy and contentment, while engagement stresses the importance of deep absorption in activities, invoking a state of flow. Positive relationships highlight the necessity of nurturing supportive and enriching connections. Meaning pertains to pursuing a purpose larger than oneself, and accomplishment encompasses striving for mastery and achievement. For the elderly, these dimensions provide a comprehensive blueprint for flourishing, guiding interventions and practices aimed at enhancing life quality and fulfilment. Well-being theory thus stands as a foundational pillar in positive psychology, offering a nuanced and holistic understanding of the factors that contribute to a life well-lived, irrespective of one’s age. === Psychological functioning and its role === Research conducted by Southwick and Charney (2012) underscores the integral role of psychological resilience in combatting depression—a condition prevalent among the elderly (16.1% of 65+-year-old Australians in [https://www.abs.gov.au/statistics/health/health-conditions-and-risks/health-conditions-prevalence/latest-release 2020–21]). Resilience does not merely denote bouncing back from adversity but signifies thriving amidst it. The elderly, with a lifetime of experiences, possess unique strengths and coping mechanisms. By harnessing these, and understanding neurobiological and psychosocial factors, there is an avenue for improved psychological functioning, ultimately contributing to enhanced flourishing. === The resilience connection === Several studies highlight that resilience is intricately linked to genetics, environment, neurobiology, and psychosocial factors (Helliwell & Sachs, 2013; Seligman, 2011; Southwick & Charney, 2012). For seniors, building resilience becomes a pivotal tool in managing age-related stressors and traumas. Embracing resilience does not mean avoiding challenges but rather developing a robust toolkit to face them head-on. With strategies encompassing positive emotions, social support, coping skills, and more, resilience offers a protective layer, allowing seniors to remain buoyant in the face of life's trials. === Proactive approaches and interventions === Promoting flourishing among the elderly necessitates actionable steps grounded in empirical evidence. Southwick and Charney (2012) advocate for interventions such as modifying the biological and psychosocial environment, strengthening social support networks, enhancing cognitive engagements, and boosting physical health through aerobic exercises. It is crucial to tailor these approaches to the distinct needs of seniors, taking into account their physiological, cognitive, and socio-emotional states. When informed by positive psychology, the amalgamation of these interventions paves the way for a life marked by heightened contentment, purpose, and joy in the latter years. Figure 3 illustrates how the integration of these supportive interventions correlates with increased levels of psychological flourishing among the elderly.<br> [[File:Psychological Flourishing - amalgamation model.png|center|thumb|400x400px|'''Figure 3'''. ''The cumulative impact of multiple interventions for supporting flourishing in the elderly''.]] == What can seniors do to flourish? == Psychological flourishing in seniors is essential for their well-being and quality of life. This section delves into practical, evidence-based strategies derived from positive psychology to foster flourishing among the elderly. It emphasises the importance of social engagement, mental stimulation, physical activity, and positive coping techniques. Tailored to individual needs, these approaches aim to enhance purpose, joy, and meaningful connections, addressing the unique challenges faced by seniors in their pursuit of a fulfilling life. === Social engagement and meaningful activities === * Building social connections rejuvenates the spirit and enhances well-being. * Social relationships have been found to significantly influence mental and emotional well-being in the elderly. A strong social network can reduce feelings of loneliness and depression, thereby promoting psychological flourishing (Cacioppo et al., 2006). * Forming bonds across different age groups can be mutually beneficial and specifically aid the elderly in feeling more connected and less isolated (Bengtson, 2001). * Engaging in community activities or volunteering (see Figure 4) has been shown to provide a sense of purpose and improve mental health outcomes for seniors (Greenfield & Marks, 2004). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': When was the last time you made a new friend? What activities can help you meet new people? :''Suggestions'': Engage in community activities, reconnect with old friends, or even consider pet ownership.<br> ::Establish regular family visits or calls, attend local gatherings, or join clubs focused on specific interests. |} [[File:Deeply engrossed in puzzle.png|thumb|317x317px|'''Figure 5'''''.'' ''An elderly woman deeply engrossed in her daily crossword puzzle; an excellent form of mental stimulation''.]] === Mental stimulation === * Continuous learning has been linked to cognitive vitality and emotional well-being (Park & Bischof, 2013). * Activities that require creativity, such as painting or music, not only stimulate the brain but also contribute to a greater sense of purpose and joy, enhancing the quality of life (Cohen et al., 2006). * Research indicates that lifelong learning and mental stimulation can help prevent cognitive decline and improve overall psychological well-being. Older adults who engage in mentally stimulating activities (see Figure 5) report higher levels of happiness and lower levels of depression (Hultsch et al., 1999). * More recent research underscores the notion that the ageing brain is capable of new neural connections when subjected to novel learning experiences, enhancing cognitive and emotional well-being (Park & Bischof, 2013). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Take up a new hobby, join an interesting class, or simply read a new book.<br> ::Explore online courses tailored for seniors or consider group-based activities to foster intellectual engagement. |} === Physical activity === [[File:TaiChi-group.png|thumb|300x280px|'''Figure 6'''''.'' ''A group of elderly enjoying a tai chi class tailored for maintaining mobility and easing arthritis symptoms''.]] * Regular physical activity boosts mental health and protects against age-related ailments such as heart disease, osteoporosis, and [https://en.wikipedia.org/wiki/Sarcopenia sarcopenia] (loss of muscle mass). * Several studies show that regular physical exercise can improve cognitive function, thereby supporting not only physical but also mental well-being (Colcombe & Kramer, 2003). * Group exercise activities like tai chi (see Figure 6) or water aerobics offer not only physical benefits but also social interaction, which can further contribute to psychological well-being (Liu & Latham, 2009). * Improved sleep through regular physical activity is correlated with better mood and mental health, providing another pathway to psychological flourishing (Reid et al., 2010). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a skill you've always wanted to learn. What's stopping you now? :''Suggestions'': Incorporate simple exercises into your daily routine, such as ::walking, yoga, or dancing, or consider joining a senior-friendly exercise group.<br> :''Please note:'' Always consult your healthcare professional before starting any new exercise regimen. |} [[File:Meditating man-solo.png|thumb|317x317px|'''Figure 7'''''.'' ''An elderly man practicing guided meditation; an excellent positive coping strategy''.]] === Positive coping strategies === * Develop resilience against challenges by adopting positive coping mechanisms such as meditation (see Figure 7), relaxation techniques, and engaging in spirituality. * Positive psychology interventions focus on strengths and virtues and have shown efficacy in improving well-being and reducing depressive symptoms in older adults (Seligman et al., 2005). * Incorporating gratitude into daily routines has been associated with positive emotional states, greater well-being, and better physical health in older adults (Emmons & McCullough, 2003). * Utilising humor is shown to not only uplift mood but also serve as an effective coping strategy for stress and life challenges. This is particularly relevant for elderly individuals who may face various forms of age-related adversity (Martin et al., 1993). {| style="border: 1px solid #A0A0A0; border-radius: 12px; display: inline-table; background-color: transparent;padding-right: 20px;" |[[File:Crystal Clear app ktip.svg|left|30px|]]'''''Consider''''': Reflect on a past challenge. How did you cope? How can you refine this strategy? :''Suggestions'': Embrace activities that foster mindfulness and reflection, such as ::meditation, gratitude journaling, or joining a support group.<br> |} {{RoundBoxTop|theme=13}} [[File:Elderly man-zest for life-mentoring.png|thumb|327x182px|'''Figure 8'''. ''John rediscovered his zest for life through mentoring teenagers, a program organised by his local APEX club''.]] '''''Can you relate''''' ... John, at 80, often sat in his armchair, lost in memories of youthful adventures. He believed his golden years had long passed, with each day echoing the sentiments of a vibrant past. Collecting his mail one day, a flyer for the local book club caught his attention, hinting at a promise of engaging discussions. Deciding to join, John discovered that the club was more than just about books; it was a community bridging generational gaps through shared stories. A young woman from the group, impressed by John's vast life experiences, introduced him to a mentoring program hosted by the local [[w:Apex Clubs of Australia|APEX]] club. Through mentoring, John shared his life lessons, offering wisdom and guidance to the younger generation. This exchange rekindled his understanding of the value of his own journey. Far from feeling that his best years were behind him, the interactions brought about a renewed sense of purpose (see Figure 8). Through the book club and mentoring, John not only found a revived passion for literature but also tapped into a deeper zest for life, seeing his age not as a limitation but as a testament to a life rich with experiences. {{RoundBoxBottom}} ==Conclusion== Ageing, despite its inherent challenges, can still be a period rich in growth, connection, and profound meaning. By integrating the principles of positive psychology with tailored interventions, an environment conducive to psychological flourishing in the elderly can be cultivated. Foundational theories such as resilience theory, self-determination theory, and well-being theory offer a robust framework for understanding and nurturing the various facets of flourishing. Translating these theories into practical strategies and interventions empowers seniors to gracefully navigate the complexities of ageing, maintaining resilience and a sustained sense of well-being. Supporting flourishing in the elderly necessitates a commitment to social engagement, enhancing emotional well-being; continuous mental stimulation, promoting cognitive vitality; regular physical activity, supporting both physical and mental health; and the adoption of positive coping strategies to foster resilience. These strategies not only enhance their quality of life but also contribute positively to the broader community, as the elderly impart their wisdom, experience, and emotional stability. Furthermore, dispelling misconceptions surrounding ageing and psychological flourishing is paramount. Challenging societal stereotypes and adopting a holistic view of ageing is essential, acknowledging the potential for growth, development, and fulfilment in this life stage. In conclusion, supporting psychological flourishing in the elderly is a multifaceted endeavour. It demands a collaborative effort from individuals, communities, and society at large, aiming to create environments that nurture the emotional, cognitive, and physical aspects of well-being. In doing so, the invaluable contributions of the elderly are honoured, fostering a culture of respect, appreciation, and comprehensive support, ensuring that the senior years are indeed filled with growth, connection, and a profound sense of purpose. ==See also== # [[Motivation and emotion/Book/2014/Ageing and emotion|Ageing and emotion]] (Book chapter, 2014) # [[Motivation and emotion/Book/2023/Ageing and motivation|Ageing and motivation]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Nudge_theory_and_sedentary_behaviour|Nudge theory and sedentary behaviour]] (Book chapter, 2023) # [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) # [[Motivation and emotion/Book/2023/Death_and_meaning_in_life|Death and meaning in life]] (Book chapter, 2023) # [[Motivation and emotion/Book/2023/Community_resilience|Community resilience]] (Book chapter, 2023) == References == {{Hanging indent|1= Ball, K., Berch, D., Helmers, K., Jobe, J., Leveck, M., Marsiske, M., Morris, J., Rebok, G., Smith, D., Tennstedt, S., Unverzagt, F., & Willis, S. (2002). Effects of cognitive training interventions with older adults. ''JAMA'', ''288''(18), 2271–2281. https://doi.org/10.1001/jama.288.18.2271 Bengtson, V. L. (2001). Beyond the nuclear family: The increasing importance of multigenerational bonds. ''Journal of Marriage and Family'', ''63''(1), 1–16. https://doi.org/10.1111/j.1741-3737.2001.00001.x Bonanno, G. A., Wortman, C. B., & Nesse, R. M. (2004). Prospective patterns of resilience and maladjustment during widowhood. ''Psychology and Aging'', ''19''(2), 260–271. https://doi.org/10.1037/0882-7974.19.2.260 Cacioppo, J. T., Hughes, M. E., Waite, L. J., Hawkley, L. C., & Thisted, R. A. (2006). Loneliness as a specific risk factor for depressive symptoms: Cross-sectional and longitudinal analyses. ''Psychology and Aging'', ''21''(1), 140–151. https://doi.org/10.1037/0882-7974.21.1.140 Carstensen, L. L., Isaacowitz, D. M., & Charles, S. T. (1999). Taking time seriously: A theory of socioemotional selectivity. ''American Psychologist'', ''54''(3), 165–181. https://doi.org/10.1037/0003-066X.54.3.165 Charles, S. T., & Carstensen, L. L. (2010). Social and emotional aging. ''Annual Review of Psychology'', ''61'', 383–409. https://doi.org/10.1146/annurev.psych.093008.100448 Chodzko-Zajko, W. J., Proctor, D. N., Fiatarone Singh, M. A., Minson, C. T., Nigg, C. R., Salem, G. J., & Skinner, J. S. (2009). Exercise and physical activity for older adults. ''Medicine & Science in Sports & Exercise'', ''41''(7), 1510–1530. https://doi.org/10.1249/MSS.0b013e3181a0c95c Cohen, G. D., Perlstein, S., Chapline, J., Kelly, J., Firth, K. M., & Simmens, S. (2006). The impact of professionally conducted cultural programs on the physical health, mental health, and social functioning of older adults. ''The Gerontologist'', ''46''(6), 726–734. https://doi.org/10.1093/geront/46.6.726 Colcombe, S., & Kramer, A. F. (2003). Fitness effects on the cognitive function of older adults: A meta-analytic study. ''Psychological Science'', ''14''(2), 125–130. https://doi.org/10.1111/1467-9280.t01-1-01430 Emmons, R. A., & McCullough, M. E. (2003). Counting blessings versus burdens: An experimental investigation of gratitude and subjective well-being in daily life. ''Journal of Personality and Social Psychology'', ''84''(2), 377–389. https://doi.org/10.1037/0022-3514.84.2.377 Greenfield, E. A., & Marks, N. F. (2004). Formal volunteering as a protective factor for older adults' psychological well-being. ''The Journals of Gerontology Series B: Psychological Sciences and Social Sciences'', ''59''(5), S258–S264. https://doi.org/10.1093/geronb/59.5.S258 Helliwell, J. F., Layard, R., & Sachs, J. (2013). World happiness report. ''United Nations Sustainable Development Solutions Network''. http://eprints.lse.ac.uk/47487/ Hultsch, D. F., Hertzog, C., Small, B. J., & Dixon, R. A. (1999). Use it or lose it: Engaged lifestyle as a buffer of cognitive decline in aging? ''Psychology and Aging'', ''14''(2), 245–263. https://doi.org/10.1037/0882-7974.14.2.245 Keyes, C. L. M. (2007). Promoting and protecting mental health as flourishing: A complementary strategy for improving national mental health. ''American Psychologist'', ''62''(2), 95–108. https://doi.org/10.1037/0003-066X.62.2.95 Liu, C. J., & Latham, N. K. (2009). Progressive resistance strength training for improving physical function in older adults. ''The Cochrane Database of Systematic Reviews'', (3), CD002759. https://doi.org/10.1002/14651858.CD002759.pub2 Martin, R. A., Kuiper, N. A., Olinger, L. J., & Dance, K. A. (1993). Humor, coping with stress, self-concept, and psychological well-being. ''Humor-International Journal of Humor Research'', ''6''(1), 89–104. https://doi.org/10.1515/humr.1993.6.1.89 Park, D. C., & Bischof, G. N. (2013). The aging mind: neuroplasticity in response to cognitive training. ''Dialogues in Clinical Neuroscience'', ''15''(1), 109–119. https://doi.org/10.31887/DCNS.2013.15.1/dpark Park, D. C., Lautenschlager, G., Hedden, T., Davidson, N. S., Smith, A. D., & Smith, P. K. (2002). Models of visuospatial and verbal memory across the adult life span. ''Psychology and Aging'', ''17''(2), 299–320. https://doi.org/10.1037/0882-7974.17.2.299 Paterson, D. H., & Warburton, D. E. (2010). Physical activity and functional limitations in older adults: a systematic review related to Canada's physical activity guidelines. ''International Journal of Behavioral Nutrition and Physical Activity'', ''7''(1), 1–22. https://doi.org/10.1186/1479-5868-7-38 Peterson, C., Ruch, W., Beermann, U., Park, N., & Seligman, M. E. P. (2007). Strengths of character, orientations to happiness, and life satisfaction. ''The Journal of Positive Psychology'', ''2''(3) 149–156. https://doi.org/10.1080/17439760701228938 Reid, K. J., Baron, K. G., Lu, B., Naylor, E., Wolfe, L., & Zee, P. C. (2010). Aerobic exercise improves self-reported sleep and quality of life in older adults with insomnia. ''Sleep Medicine'', ''11''(9), 934–940. https://doi.org/10.1016/j.sleep.2010.04.014 Ryan, R. M., & Deci, E. L. (2018). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. The Guildford Press. Seligman, M. E. P. (2011). ''Flourish''. William Heinnemann. Seligman, M. E. P., Steen, T. A., Park, N., & Peterson, C. (2005). Positive psychology progress: Empirical validation of interventions. ''American Psychologist'', ''60''(5), 410. https://doi.org/10.1037/0003-066X.60.5.410 Sin, N. L., & Lyubomirsky, S. (2009). Enhancing well-being and alleviating depressive symptoms with positive psychology interventions: A practice-friendly meta-analysis. ''Journal of Clinical Psychology'', ''65''(5), 467–487. https://doi.org/10.1002/jclp.20593 Southwick, S. M., & Charney, D. S. (2012) The science of resilience: Implications for the prevention and treatment of depression. ''Science'', ''338''(6103), 79–82. https://doi.org/10.1126/science.1222942 }} == External links == # [https://youtu.be/Is6WvYAM3gg?si=KWHkkgmP47QQDf0O 100-year olds' guide to living your best life] (Allure; YouTube) # [https://www.bluezones.com/2016/11/power-9/ Blue zones power 9: Lifestyle habits of the world’s healthiest, longest-lived people] (bluezones.com) # [https://ppc.sas.upenn.edu/ Positive psychology center] (University of Pennsylvania) # [https://youtu.be/bPBJJ-lxsXA?si=f2yDgG9X6lqiw4Cp The secret to successful aging] (Cathleen Toomey; TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:{{#titleparts:{{PAGENAME}}|3}}/Top]] [[Category:Motivation and emotion/Book/Ageing]] [[Category:Motivation and emotion/Book/Flourishing]] g7b0va7o868lkcb91e7vod62s4vazkd Is bitcoin a good investment? 0 304908 2803391 2718590 2026-04-07T19:42:41Z ~2026-21318-18 3064398 Objection to older Bitcoin claim 2803391 wikitext text/x-wiki {{Wikidebate}} Disclaimer: This is an amateur-made debate/argument analysis, not an investment advice. There is no guarantee that gaps in the arguments made will be properly pointed out in objections. ==Bitcoin is a good investment== ===Pro=== * {{Argument for}} Bitcoin is a digital analogue of gold, and in the phase before it reaches something like its saturation, it is likely to offer extreme profit yields, which, by the mean value of the investment, compensate the investor for the extreme risk. This makes bitcoin good as an item in investment portfolio, for investors who can afford to lose the bitcoin portion of investment. ** {{Objection}} Unlike bitcoin, gold has some uses beyond being a store of value. *** {{Objection}} Sure, but the price of gold is derived from its use as a store of value rather than from the other uses; the price derived solely from the other uses would be a fraction of it. ** {{Objection}} Unlike bitcoin, gold price level is fundamentally stabilized (despite its considerable volatility), not subject to the huge appreciation observed in bitcoin. * {{Argument for}} Expanding on the above, as of March 2024, bitcoin market capitalization is about 10% that of gold.<ref>[https://cryptoslate.com/insights/bitcoin-edges-closer-to-gold-with-market-cap-nearing-10/ Bitcoin edges closer to gold with market cap nearing 10%], 25 Mar 2024, cryptoslate.com</ref> If one takes bitcoin to be an analogue of gold, that suggests good room for growth. ** {{Objection}} On the other hand, as of March 2024, bitcoin market capitalization surpassed that of silver.<ref>[https://economictimes.indiatimes.com/markets/cryptocurrency/bitcoins-market-cap-surpasses-silver-becomes-eighth-most-valuable-asset/articleshow/108439631.cms Bitcoin's market cap surpasses silver, becomes eighth most valuable asset - The Economic Times], 12 Mar 2024, economictimes.indiatimes.com</ref> It is not clear what makes bitcoin more of an analogue of gold than of silver. *** {{Objection}} Gold is the leader among the physical/material/commodity store-of-value assets (unlike silver) and bitcoin is the leader among cryptographic store-of-value assets. That makes bitcoin more like gold and less like silver. * {{Argument for}} Expanding on the above, bitcoin seems to have a unique differentiator against gold in that it seems much harder to be seized/confiscated by governments. And governments sometimes turn rogue and unjustly seize property, e.g. those run by Communists in the 20th century. As per river.com, "However, bitcoin is a uniquely seizure-resistant type of property. There is no amount of physical force or legal coercion that can transfer bitcoin from one party to another without the corresponding private keys. However, if authorities can ascertain the real-world identity of an individual and their bitcoin addresses, they can coerce that individual to divulge the private keys required to move the bitcoin. Bitcoin may be seizure-resistant, but humans are still vulnerable to physical threats, blackmail, and other forms of coercion.".<ref>[https://river.com/learn/can-bitcoin-be-seized/ Can Bitcoin Be Seized?], river.com</ref> ** {{Objection}} Seizing of bitcoin by government already did happen.<ref>[https://www.reuters.com/world/europe/german-police-seizes-217-billion-bitcoin-most-extensive-action-ever-2024-01-30/ German police seizes $2.17 billion in bitcoin in 'most extensive' action ever], reuters.com</ref> * {{Argument for}} Expanding on the above, bitcoin transactions seem to provide the benefit of anonymity: while the transaction and the amount are publically available, who made the transaction not so.<ref>[https://www.coincenter.org/education/crypto-regulation-faq/how-anonymous-is-bitcoin/ How Anonymous is Bitcoin?], coincenter.org</ref> ** {{Objection}} That does not seem to provide any benefit over directly owning gold. ** {{Objection}} Not a benefit over cash transactions (as opposed to electronic ones), which can also be anonymous. ** {{Objection}} The anonymity contributes to regulatory risk; it is part of the appeal of bitcoin to criminals. ** {{Objection}} A Wired story puts the anonymity of bitcoin into doubt.<ref>[https://www.wired.com/story/27-year-old-codebreaker-busted-myth-bitcoins-anonymity/ How a 27-Year-Old Codebreaker Busted the Myth of Bitcoin’s Anonymity], 17 Jan 2024, wired.com</ref> ** ''Comment'' The purchase of bitcoin seems not anonymous when made via an intermediary. ===Con=== * {{Argument against}} Capable investor Warren Buffett dissuades from bitcoin.<ref>[https://finance.yahoo.com/news/warren-buffett-predicts-bad-ending-125222809.html Warren Buffett Predicts ‘Bad Ending’ for Bitcoin — Is It a Doomed Investment?], finance.yahoo.com</ref><ref name=nw/> Inconclusive yet suggestive. * {{Argument against}} Microsoft co-founder and philantropist Bill Gates looks down on bitcoin.<ref>[https://www.cnbc.com/2022/06/15/bill-gates-says-crypto-and-nfts-are-based-on-greater-fool-theory.html Bill Gates says crypto and NFTs are based on 'greater fool theory'], 2022, cnbc.com</ref> Inconclusive yet suggestive. * {{Argument against}} JPMorgan Chase chairman and CEO Jamie Dimon looks down on bitcoin.<ref>[https://www.cnbc.com/2021/10/11/jpmorgan-chase-ceo-jamie-dimon-says-bitcoin-is-worthless.html JPMorgan Chase CEO Jamie Dimon says bitcoin is 'worthless'] by Taylor Locke, 11 Oct 2021, cnbc.com</ref><ref>[https://www.businessinsider.com/jamie-dimon-bitcoin-crypto-ponzi-scheme-fraud-currency-blockchain-jpmorgan-2024-4 JPMorgan's Jamie Dimon Calls Bitcoin a 'Fraud' and 'Ponzi Scheme'] by Theron Mohamed, 18 Apr 2024, businessinsider.com</ref> Inconclusive yet suggestive. * {{Argument against}} Nassim Nicholas Taleb (noted for the book ''The Black Swan'') looks down on bitcoin.<ref>[https://www.fooledbyrandomness.com/BTC-QF.pdf Bitcoin, Currencies, and Fragility] by Nassim Nicholas Taleb, fooledbyrandomness.com</ref> Inconclusive yet suggestive. * {{Argument against}} Computer scientist Nicholas Weaver is very critical of cryptocurrencies.<ref>[https://law.yale.edu/sites/default/files/area/center/isp/documents/weaver_death_of_cryptocurrency_final.pdf The Death of Cryptocurrency] by Nicholas Weaver, Dec 2022, law.yale.edu</ref><ref>[https://www.currentaffairs.org/news/2022/05/why-this-computer-scientist-says-all-cryptocurrency-should-die-in-a-fire Why This Computer Scientist Says All Cryptocurrency Should “Die in a Fire”], currentaffairs.org</ref> * {{Argument against}} A bitcoin investor is exposed to regulatory risk, e.g. that state regulators will make bitcoin illegal. * {{Argument against}} A bitcoin holder is either exposed to intermediary risk (of the intermediary company defaulting) or has to own bitcoin directly using methods liable to hardware theft, hardware loss or destruction<ref>[https://www.kaspersky.com/blog/five-threats-hardware-crypto-wallets/47971/ Hot crypto wallet, cold crypto wallet: what are they, and how are they stolen from?], Kaspersky official blog</ref>, password loss/forgetting<ref name=nw>[https://www.nerdwallet.com/article/investing/is-bitcoin-a-good-investment Is Bitcoin a Good Investment?] by Kurt Woock, 11 Mar 2024, nerdwallet.com</ref>, and computer hacking. ** {{Objection}} Other currencies, such as gold or fiat, can also be lost, stolen or destroyed. * {{Argument against}} Cryptocurrencies are vulnerable to what U.K.'s FCA calls cyber-attacks (requires clarification and detail).<ref>[https://www.fca.org.uk/investsmart/investing-crypto Investing in crypto], fca.org.uk</ref> * {{Argument against}} Bitcoin has no intrinsic yield unlike e.g. a field of wheat, a block of apartment buildings or a company held via shares. ** {{Objection}} Bitcoin does not differ from gold in that regard. The above does not detract from the possibility of bitcoin providing great yields in its early adoption phase. * {{Argument against}} Bitcoin is similar to a Ponzi scheme: the late comers pay for the huge profits of the early comers. The bitcoin investors merely speculate on whether they will be the early comers--the winners--or the late comers--losers, and they bet on being the early comers. (This is an expansion on or rephrasing of the above, bitcoin having no intrinsic yield.) ** {{Objection}} Perhaps bitcoin does have an intrinsic yield, following from its serving criminals to realize their profits. *** {{Objection}} Even that kind of yield does not provide for a huge growth of bitcoin price in the decades to come: once something like a saturation overall bitcoin price level is reached, the huge profits realized from huge mid-term growth are over. ** {{Objection}} A Ponzi scheme is a fraud; by contrast, in bitcoin, the buyers know the nature of the asset (no intrinsic yield). *** {{Objection}} Fair enough as for ''Ponzi scheme''. However, the logic of the late comers paying for the profit of the early comers holds true enough. ** {{Objection}} In many investments plans, such as shares and real estate, the early comers profit from the late comers. * {{Argument against}} Bitcoin mining is energy intensive and its energy consumption grows exponentially with use.<ref>[[Wikipedia:Environmental effects of bitcoin]]</ref> Unless the underlying proof-of-work algorithm changes (like Ethereum did), Bitcoin will eventually become unsustainable and public opinion will turn against it, causing it to sink. ** {{Objection}} It is not clear what it means for bitcoin to "sink", nor is it clear how environmentalism-related unfavorable public option will cause bitcoin to lose value since its value depends only on the opinion of those who hold it and trade in it. ** {{Objection}} The Wikipedia article used above as a reference does not contain the word "exponential" and it is therefore not clear it can substantiate that bitcoin energy consumption "grows exponentially with use". ** {{Objection}} Wikipedia, used above for reference, is not a reliable source: anything that can be sourced from Wikipedia one should be able to source directly from sources used by Wikipedia. *** {{Objection}} There are many sources for bitcoin's heavy toll on the environment – more can be found here.<ref>https://www.kialo.com/cryptocurrency-mining-is-a-waste-of-resources-333.338?path=333.0~333.167-333.338</ref> **** {{Objection}} It does not seem straightforward to find the sources in the referenced kialo.com page, but if it was, it should be reasonably easy to copy the reliable sources to the present debate. **** {{Objection}} The statement to be sourced is not "heavy toll on the environment" but rather "its energy consumption grows exponentially with use". ** {{Objection}} If the market value of one bitcoin drops significantly below the energy cost (and hardware cost, etc.) required for mining an additional bitcoin, one would think the mining will stop or greatly slow down, which will solve the environmental problem. And if the energy cost of mining an additional bitcoin will be ''exponentially'' increasing, the described condition will surely set in relatively soon. ** {{Objection}} The underlying protocol that Bitcoin's Proof Of Work relies on does not require energy to be consumed at the level that imposes an environmental risk and instead shows that the active Bitcoin miners willingly choose to scale themselves to have a higher chance of mining the next block and getting the Bitcoin reward. ** {{Objection}} This might spur investments in solar and wind energy. * {{Argument against}} Bitcoin is not really a cryptocurrency (not really a currency); it is a crypto asset. Thus, it cannot realistically replace currencies. Some supporting reasoning is given by The Conversation.<ref>[https://theconversation.com/almost-no-one-uses-bitcoin-as-currency-new-data-proves-its-actually-more-like-gambling-207909 Almost no one uses Bitcoin as currency, new data proves. It’s actually more like gambling], theconversation.com</ref>. More support is at Yermack 2015.<ref>[https://www.sciencedirect.com/science/article/abs/pii/B9780128021170000023 Is Bitcoin a Real Currency? An Economic Appraisal] by David Yermack, 2015</ref> ** {{Objection}} That seems not quite true given there are companies that accept bitcoin as a method of payment.<ref>[https://swissmoney.com/who-accepts-bitcoin-as-payment 35 Companies That Accept Bitcoin & Crypto as Payment in 2024], swissmoney.com</ref> ** {{Objection}} Bitcoin does not need to be a currency (used to pay for everyday things) to be a store-of-value asset like gold and compete with gold. * {{Argument against}} An analysis by the Bank for International Settlements (BIS) reports losses of many investors: "Data on major crypto trading platforms over August 2015–December 2022 show that, as a result, a majority of crypto app users in nearly all economies made losses on their bitcoin holdings."<ref>[https://www.bis.org/publ/bisbull69.htm Crypto shocks and retail losses], 20 Feb 2023, bis.org</ref> ** {{Objection}} The analysis seems to use asssumptions that are possibly too simplifying. * {{Argument against}} There might be better cryptocurrencies in regards to profits, energy use (i.e. less harmful for the environment), and anonymity. ==References== <references/> ==Further reading== * {{W|Bitcoin#Use for investment and status as an economic bubble}}, wikipedia.org * [https://www.forbes.com/advisor/au/investing/cryptocurrency/bitcoin-price-prediction/ Bitcoin Price Prediction – Forbes Advisor Australia], forbes.com * [https://www.brookings.edu/articles/the-brutal-truth-about-bitcoin/ The brutal truth about Bitcoin], brookings.edu * [https://www.ecb.europa.eu/press/blog/date/2024/html/ecb.blog20240222~0929f86e23.en.html ETF approval for bitcoin – the naked emperor’s new clothes], 2024, ecb.europa.eu * [https://www.youtube.com/watch?v=o7zazuy_UfI Cryptocurrencies II: Last Week Tonight with John Oliver (HBO)], 24 Apr 2023, youtube.com * [https://www.nytimes.com/2018/07/31/opinion/transaction-costs-and-tethers-why-im-a-crypto-skeptic.html Opinion | Transaction Costs and Tethers: Why I’m a Crypto Skeptic] by Paul Krugman, 31 Jul 2018, nytimes.com * [https://medium.com/@CalvinCooper/rebuttal-transaction-costs-and-tethers-why-im-not-a-crypto-skeptic-e21699889943 Rebuttal — Transaction Costs and Tethers: Why I’m Not a Crypto Skeptic] by Calvin Cooper, 2018, medium.com * [https://www.cnbc.com/2018/07/09/nobel-prize-winning-economist-joseph-stiglitz-criticizes-bitcoin.html Nobel prize-winning economist Joseph Stiglitz criticizes bitcoin] by Ali Montag, 2018, cnbc.com * [https://www.youtube.com/watch?v=vdPhj3Pw-A0 Wirtschaftsprofessor: Darum hat der Bitcoin keine Zukunft // Mission Money], 2 Mar 2021, youtube.com (in German) -- an interview with Swiss economist Thorsten Hens * [https://fred.stlouisfed.org/series/CBBTCUSD Coinbase Bitcoin], fred.stlouisfed.org -- bitcoin price development [[Category:Bitcoin]] [[Category:Investment]] cl8dnzko798vhpju9urrf2d9cj3phd5 Riemann's theorem on removable singularities 0 317513 2803552 2765500 2026-04-08T10:49:15Z Bert Niehaus 2387134 Bert Niehaus moved page [[Riemann Removability Theorem]] to [[Riemann's theorem on removable singularities]]: Riemann's theorem on removable singularities 2765500 wikitext text/x-wiki ==Statement== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> be holomorphic. Then <math>f</math> can be holomorphically extended to <math>z_0</math> if and only if there exists a neighborhood <math>U \subseteq G</math> of <math>z_0</math> such that <math>f</math> is bounded on <math>U \setminus {z_0}</math>. ==Proof== Let <math>r > 0</math> be chosen such that <math>\bar{B}_r(z_0) \subseteq U</math>, and let <math>M</math> be an upper bound for <math>f</math> on <math>U</math>. We consider the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. It is {{center|<math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n, \qquad a_n = \frac{1}{2\pi i} \int_{|w-z_0| = r} \frac{f(w)}{(w-z_0)^{n+1}}\, dw </math>}}Estimating <math>a_n</math> gives the so-called Cauchy estimates, namely {{center|<math> \begin{array}{rl} |a_n| &= \displaystyle \left|\frac{1}{2\pi i} \int_{|w-z_0| = r} \frac{f(w)}{(w-z_0)^{n+1}}\, dw\right|\\ &\le \displaystyle \frac{1}{2\pi} \int_{|w-z_0| = r} \frac{|f(w)|}{|w-z_0|^{n+1}}\, |dw|\\ &\le \displaystyle \frac{1}{2\pi} \int_{|w-z_0| = r} \frac{M}{r^{n+1}}\, |dw|\\ &= \displaystyle \frac{M}{r^n} \\ \end{array} </math>}}For <math>n < 0</math>, it follows that {{center|<math> |a_n| \le \frac{M}{r^n} = Mr^{-n} \to 0, \quad r \to 0 </math>}}Thus, <math>a_n = 0</math> for all <math>n < 0</math>, meaning we have <math>f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n</math>, and <math>f(z_0) := a_0</math> is a holomorphic extension of <math>f</math> to <math>z_0</math>. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Riemannscher Hebbarkeitssatz Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Riemannscher Hebbarkeitssatz|Riemannscher Hebbarkeitssatz]] - URL:https://de.wikiversity.org/wiki/Riemannscher Hebbarkeitssatz * Date: 11/26/2024 <span type="translate" src="Riemannscher Hebbarkeitssatz" srclang="de" date="12/26/2024" time="02:00" status="inprogress"></span> <noinclude>[[de:Riemannscher Hebbarkeitssatz]]</noinclude> [[Category:Wiki2Reveal]] ate1q4vdi5r1wdghlmr0eckf8qo6c9u 2803554 2803552 2026-04-08T10:54:16Z Bert Niehaus 2387134 /* Statement */ curly bracket wrong latex syntax 2803554 wikitext text/x-wiki == Riemann Removable Singularities Theorem == Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus \{z_0\} \to \mathbb{C}</math> be holomorphic. Then <math>f</math> can be holomorphically extended to <math>z_0</math> if and only if there exists a neighborhood <math>U \subseteq G</math> of <math>z_0</math> such that <math>f</math> is bounded on <math>U \setminus \{z_0\}</math>. ==Proof== Let <math>r > 0</math> be chosen such that <math>\bar{B}_r(z_0) \subseteq U</math>, and let <math>M</math> be an upper bound for <math>f</math> on <math>U</math>. We consider the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. It is {{center|<math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n, \qquad a_n = \frac{1}{2\pi i} \int_{|w-z_0| = r} \frac{f(w)}{(w-z_0)^{n+1}}\, dw </math>}}Estimating <math>a_n</math> gives the so-called Cauchy estimates, namely {{center|<math> \begin{array}{rl} |a_n| &= \displaystyle \left|\frac{1}{2\pi i} \int_{|w-z_0| = r} \frac{f(w)}{(w-z_0)^{n+1}}\, dw\right|\\ &\le \displaystyle \frac{1}{2\pi} \int_{|w-z_0| = r} \frac{|f(w)|}{|w-z_0|^{n+1}}\, |dw|\\ &\le \displaystyle \frac{1}{2\pi} \int_{|w-z_0| = r} \frac{M}{r^{n+1}}\, |dw|\\ &= \displaystyle \frac{M}{r^n} \\ \end{array} </math>}}For <math>n < 0</math>, it follows that {{center|<math> |a_n| \le \frac{M}{r^n} = Mr^{-n} \to 0, \quad r \to 0 </math>}}Thus, <math>a_n = 0</math> for all <math>n < 0</math>, meaning we have <math>f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n</math>, and <math>f(z_0) := a_0</math> is a holomorphic extension of <math>f</math> to <math>z_0</math>. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Riemannscher Hebbarkeitssatz Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Riemannscher Hebbarkeitssatz|Riemannscher Hebbarkeitssatz]] - URL:https://de.wikiversity.org/wiki/Riemannscher Hebbarkeitssatz * Date: 11/26/2024 <span type="translate" src="Riemannscher Hebbarkeitssatz" srclang="de" date="12/26/2024" time="02:00" status="inprogress"></span> <noinclude>[[de:Riemannscher Hebbarkeitssatz]]</noinclude> [[Category:Wiki2Reveal]] tti9sdb0veqsgn639onj5027ezxv9m8 Complex Analysis/Residue 0 317633 2803546 2765695 2026-04-08T10:38:17Z Bert Niehaus 2387134 Bert Niehaus moved page [[Complex Analysis/Residuals]] to [[Complex Analysis/Residue]]: Correct title of learning resource 2765695 wikitext text/x-wiki ==Definition== Let <math>G\subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f</math> a function that is holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f: G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex Analysis/Isolated singularity|Isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation between Residue and Laurent Series == If <math>f</math> is expressed as a Laurent series around an isolated singularity <math>z_0 \in S \subset G</math>, the residue can be computed as follows: With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Complex Analysis/Laurent Expansion|Laurent Expansion]] of <math>f</math> around <math>z_0</math>, it holds: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. Here, it is assumed that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S</math>, i.e., <math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the ''residue'' <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be identified as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series of <math>f</math> around <math>z_0</math>. == Terminology == The residue (from Latin ''residuere'' - to remain) is named so because, during integration along the path <math>\gamma(t):= z_0 + r\cdot e^{it}</math> with <math>t \in [0,2\pi]</math> around <math>z_0</math>, the following holds: {{center top}} <math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1} \end{array} </math> {{center bottom}} The residue is, therefore, what "remains" after integration. == Calculation for Poles == If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Complex Analysis/Laurent Expansion|Laurent Expansion]] of <math>f</math> around <math>z_0</math> has the form: :<math> f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k </math> with <math>a_{-m} \neq 0</math>. === Proof 1: Removing the principal part by multiplication === By multiplying with <math>(z-z_0)^m</math>, we obtain: :<math> g_m(z):=(z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k </math> The residue <math>a_{-1}</math> is now the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. === Proof 2: Using (m-1)-fold differentiation === Through <math>m-1</math>-fold differentiation, the first <math>m-1</math> terms in the series, from <math>n=0</math> to <math>m-2</math>, vanish. The residue is then the coefficient of <math>(z-z_0)^0</math>, yielding: :<math> g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1} </math>. === Proof 3: Limit process to find the coefficient of <math>(z-z_0)^0</math> === By shifting the index, we obtain: :<math> g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{m+k!}{(k+1)!} a_{k}(z-z_0)^{k+1} </math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, yielding: :<math> \lim_{z\to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^{0} = (m-1)! \cdot a_{-1}</math>. Thus, the residue can be computed using the limit <math>z \to z_0</math>: :<math> \mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z\to z_0} g_m^{(m-1)}(z)</math>. == Tasks for Students == *Explain why, during integration of the Laurent series, all terms from the regular part and all terms with index <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math> contribute :<math>\int_{\partial D_r(z_0)} a_{n}\cdot (\xi - z_0)^{n}, d\xi = 0</math>. *Why is it allowed to interchange the processes of integration and series expansion? :<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d \xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d \xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f) </math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math> z \mapsto f(z)=\frac{e^{z-i}}{(z-i)^5}</math>, compute the residue <math>\mathrm{res}_{z_0}(f)</math> with <math>z_0:=i</math>!. == See Also == *[[Complex Analysis/Residuals|Residuals]] *[[w:en:Laurent series examples|Laurent series examples]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Residuals https://en.wikiversity.org/wiki/Complex%20Analysis/Residuals] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Residuals This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Residuals * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/30/2024 <span type="translate" src="Kurs:Funktionentheorie/Residuum" srclang="de" date="12/30/2024" time="12:18" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum]] </noinclude> [[Category:Wiki2Reveal]] [[Category:Complex Analysis]] ajd3nq7s4fi2g2cvvuf1w6tzo5nt8xu Complex Analysis/Decomposition theorem 0 317682 2803542 2694698 2026-04-08T10:36:28Z Bert Niehaus 2387134 2803542 wikitext text/x-wiki ==Introduction== [[File:Kreisring r R Definition.svg|thumb|Definition domain: Annular region with radii r and R]] Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series]]. == Fundamental Definitions == The following fundamental definitions are used in the decomposition theorem: : <math> K_r^R(z_o):= \{ z\in \mathbb{C} \, : \, 0 \leq r < |z-z_o| < R \}</math> : <math> K_r^\infty(z_o):= \{ z\in \mathbb{C} \, : \, 0 \leq r < |z-z_o| \}</math> : <math> D_r(z_o) := \{ z\in \mathbb{C} \, : \, |z-z_o| < r \} </math> : <math> \overline{D_r(z_o)} := \{ z\in \mathbb{C} \, : \, |z-z_o| \leq r \} </math> : <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> mit <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math> :<math> \int_{\partial D_r(z_o)} f(\xi)\, d\xi := \int_{\gamma_{r,z_0}} f(\xi)\, d\xi</math> == Decomposition Theorem for Annular Regions == Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = \{ z\in \mathbb{C} \, : \, r \leq |z-z_o| \leq R \} \subset G</math> . Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>. == Proof == The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>.By suitable composition with a shift, the decomposition theorem can be generalized<math>K_r^R(z_o) = \{ z\in \mathbb{C} \, : \, r \leq |z-z_o| \leq R \}</math> for arbitrary with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed. ===Proof Idea=== *The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>. *Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation. *Application of the Cauchy Integral Theorem for cycles. *Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries. *One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral. ===Uniqueness of the Decomposition=== The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder. ===Proof 1: Circular Rings with Center 0=== We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> with <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with: : <math>G_{z_o} = \{ z-z_o \in \mathbb{C} \, : \, z \in G \}</math> mit <math>0 \in G_{z_o}</math> da <math>z_o \in G</math> : <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> mit <math> f_{z_o}(z) := f(z-z_o)</math> : <math>\overline{K_r^R(0)} \subset G_{z_o}</math> da <math>\overline{K_r^R(z_o)} \subset G</math> ===Proof 2: Definition of the Boundary Cycle for the Circular Ring=== We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations. : <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>. : <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>. : <math> \Gamma := \gamma_1 + \gamma_2</math>. ==Proof 3: Null-Homologous Cycle== Since<math>\overline{K_r^R(z_o)} = \{ z\in \mathbb{C} \, : \, r \leq |z-z_o| \leq R \} \subset G</math> is <math>\Gamma</math> a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>. ===Proof 4: Application of the Cauchy Integral Formula for Cycles=== For <math>z \in K_r^R(z_o)</math>, the following holds: :<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>. === Proof 5: Substitution for Integration Paths === Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths: :<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math> The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>. === Proof 6: Standard Estimation === We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math> :<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math> :: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math> Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted. === Proof 7: Limit Process for Standard Estimation === Thus, we obtain: :: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits_{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math> === Proof 8: Taylor Expansion with Cauchy Kernel === The series expansion occurs with the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] in the red-marked convergence area: :<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}</math> (see also [[Abel's Lemma]]). === Proof 9: Extension of the Domain to the Interior of the Annulus === The domains of the two functions are currently limited to the annular region. [[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]] We extend the function <math>f_1</math> step by step to the interior of the annulus. === Proof 10: Development Point Moves to the Inner Circle === The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series. Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region. === Proof 11: Moving Circular Regions as Convergence Region for Power Series === [[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]] === Proof 11: Holomorphic Extension - Disk, Annulus === <p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p> [[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]] <p> The red annulus represents the extension with all disks and development points on the trace. </p> === Proof 12: Convergence Region of the Power Series === For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem. === Proofs 13: Iteratively Extend the Convergence Region === Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>. [[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]] The development points of the Taylor expansions now lie along a circular integration path with a smaller radius. === Proof 14a: Transformation of Domain for the Main Part === For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>: :<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math> With this transformation, we have: :<math>|z| < \frac{1}{r} \, \Longrightarrow |T(z)-z_o| > r </math> Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0\in\mathbb{C}</math>. However, <math>0\in\mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>. === Proof 14b: Analogous Approach for the Main Part === Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined. == See also == *[[Abel's Lemma]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/decomposition%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=decomposition%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/decomposition%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=decomposition%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/decomposition%20theorem https://en.wikiversity.org/wiki/Complex%20Analysis/decomposition%20theorem] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/decomposition%20theorem This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/decomposition%20theorem * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/decomposition%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=decomposition%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zerlegungssatz Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Zerlegungssatz|Kurs:Funktionentheorie/Zerlegungssatz]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zerlegungssatz * Date: 1/2/2025 <span type="translate" src="Kurs:Funktionentheorie/Zerlegungssatz" srclang="de" date="1/2/2025" time="11:12" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Zerlegungssatz]] </noinclude> [[Category:Wiki2Reveal]] [[Category:Complex Analysis]] 12gcysd04qgwm60ef38h11lptvccs6a 2803543 2803542 2026-04-08T10:37:07Z Bert Niehaus 2387134 Bert Niehaus moved page [[Complex Analysis/decomposition theorem]] to [[Complex Analysis/Decomposition theorem]]: capitalize first character 2803542 wikitext text/x-wiki ==Introduction== [[File:Kreisring r R Definition.svg|thumb|Definition domain: Annular region with radii r and R]] Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series]]. == Fundamental Definitions == The following fundamental definitions are used in the decomposition theorem: : <math> K_r^R(z_o):= \{ z\in \mathbb{C} \, : \, 0 \leq r < |z-z_o| < R \}</math> : <math> K_r^\infty(z_o):= \{ z\in \mathbb{C} \, : \, 0 \leq r < |z-z_o| \}</math> : <math> D_r(z_o) := \{ z\in \mathbb{C} \, : \, |z-z_o| < r \} </math> : <math> \overline{D_r(z_o)} := \{ z\in \mathbb{C} \, : \, |z-z_o| \leq r \} </math> : <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> mit <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math> :<math> \int_{\partial D_r(z_o)} f(\xi)\, d\xi := \int_{\gamma_{r,z_0}} f(\xi)\, d\xi</math> == Decomposition Theorem for Annular Regions == Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = \{ z\in \mathbb{C} \, : \, r \leq |z-z_o| \leq R \} \subset G</math> . Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>. == Proof == The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>.By suitable composition with a shift, the decomposition theorem can be generalized<math>K_r^R(z_o) = \{ z\in \mathbb{C} \, : \, r \leq |z-z_o| \leq R \}</math> for arbitrary with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed. ===Proof Idea=== *The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>. *Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation. *Application of the Cauchy Integral Theorem for cycles. *Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries. *One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral. ===Uniqueness of the Decomposition=== The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder. ===Proof 1: Circular Rings with Center 0=== We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> with <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with: : <math>G_{z_o} = \{ z-z_o \in \mathbb{C} \, : \, z \in G \}</math> mit <math>0 \in G_{z_o}</math> da <math>z_o \in G</math> : <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> mit <math> f_{z_o}(z) := f(z-z_o)</math> : <math>\overline{K_r^R(0)} \subset G_{z_o}</math> da <math>\overline{K_r^R(z_o)} \subset G</math> ===Proof 2: Definition of the Boundary Cycle for the Circular Ring=== We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations. : <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>. : <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>. : <math> \Gamma := \gamma_1 + \gamma_2</math>. ==Proof 3: Null-Homologous Cycle== Since<math>\overline{K_r^R(z_o)} = \{ z\in \mathbb{C} \, : \, r \leq |z-z_o| \leq R \} \subset G</math> is <math>\Gamma</math> a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>. ===Proof 4: Application of the Cauchy Integral Formula for Cycles=== For <math>z \in K_r^R(z_o)</math>, the following holds: :<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>. === Proof 5: Substitution for Integration Paths === Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths: :<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math> The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>. === Proof 6: Standard Estimation === We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math> :<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math> :: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math> Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted. === Proof 7: Limit Process for Standard Estimation === Thus, we obtain: :: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits_{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math> === Proof 8: Taylor Expansion with Cauchy Kernel === The series expansion occurs with the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] in the red-marked convergence area: :<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}</math> (see also [[Abel's Lemma]]). === Proof 9: Extension of the Domain to the Interior of the Annulus === The domains of the two functions are currently limited to the annular region. [[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]] We extend the function <math>f_1</math> step by step to the interior of the annulus. === Proof 10: Development Point Moves to the Inner Circle === The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series. Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region. === Proof 11: Moving Circular Regions as Convergence Region for Power Series === [[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]] === Proof 11: Holomorphic Extension - Disk, Annulus === <p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p> [[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]] <p> The red annulus represents the extension with all disks and development points on the trace. </p> === Proof 12: Convergence Region of the Power Series === For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem. === Proofs 13: Iteratively Extend the Convergence Region === Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>. [[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]] The development points of the Taylor expansions now lie along a circular integration path with a smaller radius. === Proof 14a: Transformation of Domain for the Main Part === For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>: :<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math> With this transformation, we have: :<math>|z| < \frac{1}{r} \, \Longrightarrow |T(z)-z_o| > r </math> Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0\in\mathbb{C}</math>. However, <math>0\in\mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>. === Proof 14b: Analogous Approach for the Main Part === Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined. == See also == *[[Abel's Lemma]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/decomposition%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=decomposition%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/decomposition%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=decomposition%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/decomposition%20theorem https://en.wikiversity.org/wiki/Complex%20Analysis/decomposition%20theorem] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/decomposition%20theorem This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/decomposition%20theorem * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/decomposition%20theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=decomposition%20theorem&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zerlegungssatz Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Zerlegungssatz|Kurs:Funktionentheorie/Zerlegungssatz]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zerlegungssatz * Date: 1/2/2025 <span type="translate" src="Kurs:Funktionentheorie/Zerlegungssatz" srclang="de" date="1/2/2025" time="11:12" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Zerlegungssatz]] </noinclude> [[Category:Wiki2Reveal]] [[Category:Complex Analysis]] 12gcysd04qgwm60ef38h11lptvccs6a Universal Bibliography/Music 0 321521 2803419 2802859 2026-04-07T21:04:04Z James500 297601 Add 2803419 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].] *[http://music-calendar.jp/ Music Calendar] Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] **Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208. *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:Oricon|Oricon]] (オリコン) *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). DJs *Masahiro Yasuda, "How Japanese DJs cut across Market Boundaries" (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45 [[Category:Music]] kiel99yzxbb1umrrwycdqmfvdvnm4xo 2803423 2803419 2026-04-07T21:06:08Z James500 297601 Not needed 2803423 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].] *[http://music-calendar.jp Music Calendar] Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] **Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208. *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:Oricon|Oricon]] (オリコン) *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). DJs *Masahiro Yasuda, "How Japanese DJs cut across Market Boundaries" (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45 [[Category:Music]] q1rturaul6kzkj7hi3n717rupp3qg0t 2803438 2803423 2026-04-07T21:43:41Z James500 297601 Add 2803438 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].] *[http://music-calendar.jp Music Calendar] Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Homogeneity of pop music: *[https://www.theguardian.com/music/2012/jul/27/pop-music-sounds-same-survey-reveals Pop music these days: it all sounds the same, survey reveals]. The Guardian. 27 July 2012. *[https://www.nbcnews.com/id/wbna48356108 Pop Music All Sounds the Same Nowadays]. NBC News. 27 July 2012. *[https://www.independent.co.uk/voices/comment/why-does-today-s-pop-music-sound-the-same-because-the-same-people-make-it-8368714.html Why does today's pop music sound the same? Because the same people make it]. The Independent. 29 November 2012. *[https://www.reuters.com/article/lifestyle/science/pop-music-too-loud-and-all-sounds-the-same-official-idUSBRE86P0R9/ Pop music too loud and all sounds the same: official]. Reuters. 26 July 2012. *[https://theconversation.com/from-art-form-to-asset-our-study-found-popular-songs-are-becoming-more-generic-266097 From art form to asset: our study found popular songs are becoming more generic]. The Conversation. 3 October 2025. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] **Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208. *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:Oricon|Oricon]] (オリコン) *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). DJs *Masahiro Yasuda, "How Japanese DJs cut across Market Boundaries" (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45 [[Category:Music]] 5p4nv55rrlta0gau5tlhaczwnmqmuvy Just sustainability transitions: a living review 0 326060 2803329 2798674 2026-04-07T13:41:27Z Jeanne Noiraud 1366702 /* Article classification */ 2803329 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification === Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} ebc3shp8ywcroa17a7hxxwt65llg7iz 2803330 2803329 2026-04-07T13:46:57Z Jeanne Noiraud 1366702 /* Methodology */ 2803330 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner --> === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} i7ok9zifrtj1tp3abjokiymeecuycmj 2803331 2803330 2026-04-07T13:48:03Z Jeanne Noiraud 1366702 /* Methodology */ 2803331 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner --> === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : <!-- include all below items using the wikidata link template --> Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <!-- include all below items using the wikidata link template --> <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 98sygs2hw7suqvx1n9ha2yw1n2nozyd 2803523 2803331 2026-04-08T09:31:56Z Amélie E. Pereira 3042711 /* Thematic networks */ + bugs 2803523 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner --> === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : <!-- include all below items using the wikidata link template --> Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <!-- include all below items using the wikidata link template --> <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. During the construction of the graph, the main difficulty encountered was the numerous bugs in the open-source tools used. For example, the "Author Disambiguator" tool, used to create entries for the researchers who worked on the publications we are analysing, fails to launch about half the time, displaying the message "too many requests". We asked the Northwestern Europe hackathon to fix the bug relating to the Cita tool’s connection. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 8oei9hty8y1imge7s74wjp8hsh0215t 2803524 2803523 2026-04-08T09:36:08Z Amélie E. Pereira 3042711 /* Difficulties encountered in modelling concepts: */ 2803524 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner --> === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : <!-- include all below items using the wikidata link template --> Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <!-- include all below items using the wikidata link template --> <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. During the construction of the graph, the main difficulty encountered was the numerous bugs in the open-source tools used. For example, the "Author Disambiguator" tool, used to create entries for the researchers who worked on the publications we are analysing, fails to launch about half the time, displaying the message "too many requests". We asked the Northwestern Europe hackathon to fix the bug relating to the Cita tool’s connection. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals The Orcidator tool, which, as its name suggests, automatically adds ORCIDs to researchers’ profiles, could not be used. The message "DEACTIVATED BECAUSE OF ABUSE" appears after logging in (https://sourcemd.toolforge.org/orcidator_old.php, tested on 7 April 2026) ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} htfxyjcvg2199pf6nnz81lss7hwzg9d 2803525 2803524 2026-04-08T09:37:31Z Amélie E. Pereira 3042711 /* Knowledge modelling */ 2803525 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner --> === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : <!-- include all below items using the wikidata link template --> Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <!-- include all below items using the wikidata link template --> <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals During the construction of the graph, the main difficulty encountered was the numerous bugs in the open-source tools used. For example, the "Author Disambiguator" tool, used to create entries for the researchers who worked on the publications we are analysing, fails to launch about half the time, displaying the message "too many requests" The Orcidator tool, which, as its name suggests, automatically adds ORCIDs to researchers’ profiles, could not be used. The message "DEACTIVATED BECAUSE OF ABUSE" appears after logging<ref>https://sourcemd.toolforge.org/orcidator_old.php, tested on 7 April 2026</ref>. ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 742rwdo2w9mpf8k6oklm8rqxcwameyo 2803528 2803525 2026-04-08T09:41:39Z Amélie E. Pereira 3042711 /* Knowledge modelling */ 2803528 wikitext text/x-wiki == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner --> === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : <!-- include all below items using the wikidata link template --> Q42377797 <nowiki>https://www.wikidata.org/wiki/Q2798912</nowiki> <nowiki>https://www.wikidata.org/wiki/Q421953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q84459973</nowiki> <nowiki>https://www.wikidata.org/wiki/Q185836</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4764988</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4338318</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4930066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q430460</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7569</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4116870</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125928</nowiki> <nowiki>https://www.wikidata.org/wiki/Q260607</nowiki> <nowiki>https://en.wikipedia.org/wiki/Climate_change_mitigation</nowiki> Q1291678 Q2270945 <nowiki>https://www.wikidata.org/wiki/Q16972712</nowiki> Q16324410 <nowiki>https://www.wikidata.org/wiki/Q11024</nowiki> <nowiki>https://www.wikidata.org/wiki/Q177634</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5154673</nowiki> Q113514984 <nowiki>https://www.wikidata.org/wiki/Q65807646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188843</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11693783</nowiki> <nowiki>https://www.wikidata.org/wiki/Q284289</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7174</nowiki> Q552284 <nowiki>https://www.wikidata.org/wiki/Q1230584</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1049066</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8134</nowiki> <nowiki>https://www.wikidata.org/wiki/Q295865</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1358789</nowiki> <nowiki>https://www.wikidata.org/wiki/Q868575</nowiki> <nowiki>https://www.wikidata.org/wiki/Q138359220</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131444737</nowiki> www.wikidata.org/wiki/Q16869822 Q14944319 <nowiki>https://www.wikidata.org/wiki/Q192704</nowiki> Q117091181 <nowiki>https://www.wikidata.org/wiki/Q24965464</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1805337</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1341244</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3406659</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3456219</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2700433</nowiki> <nowiki>https://www.wikidata.org/wiki/Q837718</nowiki> Q795757 Q795757 Q1479527 <nowiki>https://www.wikidata.org/wiki/Q771773</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56395513</nowiki> <nowiki>https://www.wikidata.org/wiki/Q5465532</nowiki> <nowiki>https://www.wikidata.org/wiki/Q4421</nowiki> <nowiki>https://www.wikidata.org/wiki/Q48277</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1553864</nowiki> <nowiki>https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0</nowiki> <nowiki>https://www.wikidata.org/wiki/Q11376059</nowiki> <nowiki>https://www.wikidata.org/wiki/Q103817</nowiki> <nowiki>https://www.wikidata.org/wiki/Q113561794</nowiki> <nowiki>https://www.wikidata.org/wiki/Q770480</nowiki> Q17142211 <nowiki>https://www.wikidata.org/wiki/Q1516555</nowiki> Q6316391 <nowiki>https://www.wikidata.org/wiki/Q366139</nowiki> Q3027857 <nowiki>https://www.wikidata.org/wiki/Q59679511</nowiki> <nowiki>https://www.wikidata.org/wiki/Q43619</nowiki> <nowiki>https://www.wikidata.org/wiki/Q127514833</nowiki> <nowiki>https://www.wikidata.org/wiki/Q13023682</nowiki> <nowiki>https://www.wikidata.org/wiki/Q728646</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3907287</nowiki> <nowiki>https://www.wikidata.org/wiki/Q9357091</nowiki> <nowiki>https://www.wikidata.org/wiki/Q265425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q25107</nowiki> Q442100 <nowiki>https://www.wikidata.org/wiki/Q7249406</nowiki> Q7257735 <nowiki>https://www.wikidata.org/wiki/Q541936</nowiki> Q6142016 <nowiki>https://www.wikidata.org/wiki/Q10509953</nowiki> <nowiki>https://www.wikidata.org/wiki/Q12705</nowiki> <nowiki>https://www.wikidata.org/wiki/Q56510941</nowiki> Q1165392 <nowiki>https://www.wikidata.org/wiki/Q4414036</nowiki> <nowiki>https://www.wikidata.org/wiki/Q17152351</nowiki> <nowiki>https://www.wikidata.org/wiki/Q187588</nowiki> <nowiki>https://www.wikidata.org/wiki/Q264892</nowiki> <nowiki>https://www.wikidata.org/wiki/Q34749</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2930198</nowiki> <nowiki>https://www.wikidata.org/wiki/Q125359881</nowiki> <nowiki>https://www.wikidata.org/wiki/Q219416</nowiki> <nowiki>https://www.wikidata.org/wiki/Q131201</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7649586</nowiki> <nowiki>https://www.wikidata.org/wiki/Q69883</nowiki> <nowiki>https://www.wikidata.org/wiki/Q920600</nowiki> <nowiki>https://www.wikidata.org/wiki/Q3376054</nowiki> <nowiki>https://www.wikidata.org/wiki/Q107389921</nowiki> <nowiki>https://www.wikidata.org/wiki/Q7981051</nowiki> <nowiki>https://www.wikidata.org/wiki/Q467</nowiki> <nowiki>https://www.wikidata.org/wiki/Q188867</nowiki> <nowiki>https://www.wikidata.org/wiki/Q1038171</nowiki> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : <!-- include all below items using the wikidata link template --> <nowiki>https://www.wikidata.org/wiki/Q603441</nowiki> <nowiki>http://www.wikidata.org/entity/Q472342</nowiki> <nowiki>http://www.wikidata.org/entity/Q815382</nowiki> <nowiki>http://www.wikidata.org/entity/Q1504425</nowiki> <nowiki>https://www.wikidata.org/wiki/Q2412849</nowiki> <nowiki>http://www.wikidata.org/entity/Q6822263</nowiki> <nowiki>http://www.wikidata.org/entity/Q7301211</nowiki> <nowiki>http://www.wikidata.org/entity/Q17007303</nowiki> <nowiki>http://www.wikidata.org/entity/Q70470634</nowiki> <nowiki>http://www.wikidata.org/entity/Q101116078</nowiki> <nowiki>http://www.wikidata.org/entity/Q110665014</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174203</nowiki> <nowiki>http://www.wikidata.org/entity/Q137174450</nowiki> <nowiki>http://www.wikidata.org/entity/Q137209848</nowiki> <nowiki>http://www.wikidata.org/entity/Q137211242</nowiki> [Include list and description of types of litterature reviews] Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. === Knowledge modelling === Concept maps can also be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. We used the Author Disambiguator tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers: following a query, it categorises scientific publications into thematic groups. It also automatically searches for ORCID, ResearchGate and VIAF pages<ref>https://author-disambiguator.toolforge.org/</ref>. ==== Thematic networks ==== A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as grounded theory<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. [Illustration of thematic networks] Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. ==== Conceptual modelling ==== Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ===== Difficulties encountered in modelling concepts: ===== *{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent. * {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals During the construction of the graph, the main difficulty encountered was the numerous bugs in the open-source tools used. For example, the "Author Disambiguator" tool, used to create entries for the researchers who worked on the publications we are analysing, fails to launch about half the time, displaying the message "too many requests" The Orcidator tool, which, as its name suggests, automatically adds ORCIDs to researchers’ profiles, could not be used. The message "DEACTIVATED BECAUSE OF ABUSE" appears after logging<ref>https://sourcemd.toolforge.org/orcidator_old.php, tested on 7 April 2026</ref>. ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors '''Chronologies''' === Writing === To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 0ycsyw9zurdge6z3na92riro9vsbilk Patriarch Ages Curious Numerical Facts Response 0 328204 2803380 2803273 2026-04-07T18:55:40Z CanonicalMormon 2646631 /* Fathering Ages */ 2803380 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. == Bottom Line Up Front == === Death Ages === Many of the patriarchs' death ages appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative death ages of the patriarchs from Adam to Moses derive from the "perfect" Mesopotamian number of seven ''šar'' (or 420 ''šūši'') divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] ov3tyivfipj825otbrraqmcqvef3ttv 2803381 2803380 2026-04-07T18:56:31Z CanonicalMormon 2646631 /* Bottom Line Up Front */ 2803381 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' death ages appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative death ages of the patriarchs from Adam to Moses derive from the "perfect" Mesopotamian number of seven ''šar'' (or 420 ''šūši'') divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] nyzi2s1u3qec3iqu1tmwxi7x6uxp9kb 2803383 2803381 2026-04-07T18:57:45Z CanonicalMormon 2646631 /* Arichat Yamim */ 2803383 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' death ages that we find in the Hebrew bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative death ages of the patriarchs from Adam to Moses derive from the "perfect" Mesopotamian number of seven ''šar'' (or 420 ''šūši'') divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] j85ye26pb7fu1kfq15ta635u0r0ghac 2803385 2803383 2026-04-07T19:09:45Z CanonicalMormon 2646631 /* Arichat Yamim */ 2803385 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 6omkkp7y59r86ht36bto9fkptncypl8 2803386 2803385 2026-04-07T19:12:42Z CanonicalMormon 2646631 /* Prototype 2: Refined "Jubilee" Allocation */ 2803386 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 6ui290iw027j81bpv1pnbvxj4715fn7 2803387 2803386 2026-04-07T19:17:27Z CanonicalMormon 2646631 /* Edit 0 */ 2803387 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 8fu7dmdk1wqbw89lhjh8lmgz4ct1k7h 2803392 2803387 2026-04-07T19:47:33Z CanonicalMormon 2646631 /* Edit 0 */ 2803392 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] qqsn7hzxfog0j8ox93vi76vaoo2vmgr 2803394 2803392 2026-04-07T19:51:17Z CanonicalMormon 2646631 /* Edit 0 */ 2803394 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 7k08zr2c795vu6mv833xw5d8jxdz67x 2803395 2803394 2026-04-07T20:05:14Z CanonicalMormon 2646631 /* PT2 as the Base Model for Patriarchal Chronologies */ 2803395 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = Edit 0 = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] ms6mt1jocrf9p0qrwkrej6ieddvfvhd 2803396 2803395 2026-04-07T20:06:07Z CanonicalMormon 2646631 /* Edit 0 */ 2803396 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] npmjku40vbjh7cs7s0ztifrqce72eql 2803397 2803396 2026-04-07T20:06:35Z CanonicalMormon 2646631 /* PT2 as the Base Model for Patriarchal Chronologies */ 2803397 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Many of the patriarchs' lifespans that we find in the Hebrew Bible and related chronologies appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). The sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] dgcncexcchl7i2px2xyw6mg3psw8o8q 2803399 2803397 2026-04-07T20:14:25Z CanonicalMormon 2646631 /* Arichat Yamim */ 2803399 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] qn3safip2pzny8z9b3ic89icnri81t4 2803400 2803399 2026-04-07T20:19:54Z CanonicalMormon 2646631 /* PT2 as the Base Model for Patriarchal Chronologies */ 2803400 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} === PT2 as the Base Model for Patriarchal Chronologies === The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> ===Comparative Chronology Tables=== The following tables reconstruct '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] nikm1wz2cmzpbd86437qgpa5kxu26bv 2803402 2803400 2026-04-07T20:20:32Z CanonicalMormon 2646631 /* PT2 as the Base Model for Patriarchal Chronologies */ 2803402 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> ===Comparative Chronology Tables=== The following tables reconstruct '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] i3msdnrb0qtpvbudrguewj65puxdaof 2803410 2803402 2026-04-07T20:40:14Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803410 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across all other known traditions, these standard values have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] s23fi51mv7uc0ja1ejqm53sqnmefu04 2803411 2803410 2026-04-07T20:41:02Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803411 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known traditions, these standard values have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] nou22qo30bbgl4mfaw41ojulqq1rpcj 2803412 2803411 2026-04-07T20:43:10Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803412 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] o8a99mmqswmtpshcp3l9d7j7vyxu5hy 2803416 2803412 2026-04-07T20:55:05Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803416 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following comparison, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] h0gqwx3lzwjxu7rnm20pod5ssunrtvy 2803417 2803416 2026-04-07T21:01:50Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803417 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following comparison, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required lifespan reduction, to ensure that Jared died in the year of the flood, was 115 years. Surprisingly (as noted in the previous section), the Samaritans also reduced the lifespans of a few later patriarchs by 115 years to maintain a balance between the Group 1 and group 2 patriarchs. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] cn9smevnlkofb9sp4klpk459g9209u0 2803418 2803417 2026-04-07T21:02:54Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803418 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following comparison, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required lifespan reduction, to ensure that Jared died in the year of the flood, was 115 years. Surprisingly (as noted in the previous section), the Samaritans also reduced the lifespans of a few later patriarchs by a combined total of 115 years to maintain a balance between the Group 1 and Group 2 patriarchs described previously. In particular, '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 1s7by38qgbd25iutt9b0n0olmwxk3qk 2803420 2803418 2026-04-07T21:04:39Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803420 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following comparison, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required lifespan reduction, to ensure that Jared died in the year of the flood, was 115 years. Surprisingly (as noted in the previous section), the Samaritans also reduced the lifespans of a few later patriarchs by a combined total of 115 years to maintain a balance between the Group 1 and Group 2 patriarchs described previously. In particular, Eber and Terah each had their respective lifespan reduced by 60 years, or one šūši. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] thqhpz4g0cu029ex004oz4zqs47v8w2 2803421 2803420 2026-04-07T21:05:35Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803421 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required lifespan reduction, to ensure that Jared died in the year of the flood, was 115 years. Surprisingly (as noted in the previous section), the Samaritans also reduced the lifespans of a few later patriarchs by a combined total of 115 years to maintain a balance between the Group 1 and Group 2 patriarchs described previously. In particular, Eber and Terah each had their respective lifespan reduced by 60 years, or one šūši. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] nuxpkttqgr9lr39qtf1mvyejjfucbiu 2803422 2803421 2026-04-07T21:06:01Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803422 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required lifespan reduction, to ensure that Jared died in the year of the flood, was 115 years. Surprisingly (as noted in the previous section), the Samaritans also reduced the lifespans of a few later patriarchs by a combined total of 115 years to maintain a balance between the Group 1 and Group 2 patriarchs described previously. In particular, Eber and Terah each had their respective lifespan reduced by 60 years, or one šūši each. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] n75i5jl0dp9grca4l3pr8ig76fqxdxu 2803424 2803422 2026-04-07T21:09:03Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803424 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] fwiqk5pcoevej3n40vdmckigvbckggq 2803425 2803424 2026-04-07T21:10:46Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803425 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] s1tqlvd7wdq0wq3v9byjn1gwcgpazps 2803426 2803425 2026-04-07T21:11:56Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803426 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the following table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] qwuvhvdfkhrr9fx0oxg64qwrg280059 2803429 2803426 2026-04-07T21:13:58Z CanonicalMormon 2646631 /* Samaritan Adjustments */ 2803429 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] ptg29ngf2z52mmqvmz4dokfb47597l1 2803445 2803429 2026-04-08T01:30:39Z CanonicalMormon 2646631 /* Samaritan Adjustments */ 2803445 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following tables reconstruct lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts, the '''Armenian Eusebius''' chronology does not explicitly list death ages for Levi, Kohath, and Amram. Because these specific lifespans are uniform across other known ''Long Chronology'' traditions, these standard values have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === There are five different traditional values for Lamech’s death age listed in the above table: 783 (PT2), 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] dfk3c7o332mtjgdwgv9ntrvxr19n5b2 2803446 2803445 2026-04-08T01:39:00Z CanonicalMormon 2646631 /* Comparative Chronology Tables */ 2803446 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === There are five different traditional values for Lamech’s death age listed in the above table: 783 (PT2), 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] fxs0khkkhluhef2dzwly2qep1e9o3gj 2803447 2803446 2026-04-08T01:44:09Z CanonicalMormon 2646631 /* Masoretic Adjustments */ 2803447 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === There are five different values for Lamech’s death age listed in the above table: 783 (PT2), 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] cp188yw8aypiywenzh1gbgt2sm2zymw 2803448 2803447 2026-04-08T01:53:54Z CanonicalMormon 2646631 /* Masoretic Adjustments */ 2803448 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', Paul D notes the following regarding Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] qg8r00gez48qnc54oqvubqp0ifepszi 2803449 2803448 2026-04-08T02:12:56Z CanonicalMormon 2646631 /* Masoretic Adjustments */ 2803449 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', Paul D notes the following regarding Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing Applying ''Lectio Difficilior'', we should set aside the figures 53, 500, and 753 as being "too structured to be true. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts 180 years as Isaac's original life span, then another suspiciously neat pattern emerges: * Abraham's lifespan: '''175 years (7 × 5<sup>2</sup>). * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 3lkoniitirno3ki2p7kf87q7vvmc9bn 2803450 2803449 2026-04-08T02:23:16Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803450 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', Paul D notes the following regarding Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing Applying ''Lectio Difficilior'', we should set aside the figures 53, 500, and 753 as being "too structured to be true". A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts 180 years as Isaac's original life span, then another suspiciously neat pattern emerges: * Abraham's lifespan: '''175 years (7 × 5<sup>2</sup>). * Isaac's lifespan: '''180 years (5 × 6<sup>2</sup>). * Jacob's lifespan: '''147 years (3 × 7<sup>2</sup>). Once again, we should set aside these figures as being "too structured to be true". In the PT2 prototype chronology, it is assumed that Isaac's original lifespan was 174 years (a value which is preserved exclusively in the book of Jubilees. Isaac's lifespan was later increased by eight years to achieve the above mathematical relationship between Abraham and Jacob, and Lamech's lifespan was decreased by eight years to achieve the 777 figure describe previously. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] f60i0xevirqzdoi4qdl74990gzsxbh4 2803453 2803450 2026-04-08T02:35:12Z CanonicalMormon 2646631 /* Masoretic Adjustments */ 2803453 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the [[Book of Jubilees]]. Under this theory, Isaac's lifespan was later increased by six years to achieve the mathematical relationship above, while Lamech's lifespan was adjusted to achieve the '''777''' figure described previously, suggesting a deliberate "balancing" of the total chronological years across different traditions. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] sc4hx5inzajyq99t6fidt0fdyhr4d61 2803455 2803453 2026-04-08T02:41:31Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803455 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the Book of Jubilees. Under this theory, Isaac's lifespan was later increased by six years to achieve the mathematical relationship above, while Lamech's lifespan was adjusted to achieve the '''777''' figure described previously, suggesting a deliberate "balancing" of the total chronological years across different traditions. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 87vjdr5n74augy0w01u4y80kmfyl166 2803457 2803455 2026-04-08T02:45:53Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803457 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the Book of Jubilees. Under this theory, Isaac's lifespan was later increased by six years to achieve the mathematical relationship above, while Lamech's lifespan was adjusted to achieve the '''777''' figure described previously, suggesting a deliberate "balancing" of total chronological years. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in the previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] gbbfii3eqkkxw1vgbywxgfbx9hni06s 2803458 2803457 2026-04-08T02:46:54Z CanonicalMormon 2646631 /* Samaritan Adjustments */ 2803458 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the Book of Jubilees. Under this theory, Isaac's lifespan was later increased by six years to achieve the mathematical relationship above, while Lamech's lifespan was adjusted to achieve the '''777''' figure described previously, suggesting a deliberate "balancing" of total chronological years. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] b3d0dwvujzye13zphupf9q0mb99wmkd 2803461 2803458 2026-04-08T02:49:32Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803461 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the Book of Jubilees. Under this theory, Isaac's lifespan was increased by six years to achieve the mathematical relationship above, while Lamech's lifespan was adjusted to achieve the '''777''' figure described previously, suggesting a deliberate "balancing" of total chronological years. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 11acrl2f9cas4i6k0wnr0kx58yp7z5h 2803462 2803461 2026-04-08T02:58:32Z CanonicalMormon 2646631 /* Samaritan Adjustments */ 2803462 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the Book of Jubilees. Under this theory, Isaac's lifespan was increased by six years to achieve the mathematical relationship above, while Lamech's lifespan was adjusted to achieve the '''777''' figure described previously, suggesting a deliberate "balancing" of total chronological years. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. === Armenian Eusebius Adjustments === The Armenian Eusebius Chronology: This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or 10 šūši. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] ez1g8nl5dzrt7k2hulrybhexunmr53l 2803463 2803462 2026-04-08T03:08:56Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803463 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of the article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the [[Book of Jubilees]]. Under this theory, Isaac's lifespan was increased by six years in the Masoretic, Samaritan, and "Long Chronology" traditions to achieve the mathematical relationship noted above. Conversely, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously. This suggests a deliberate "balancing" of total chronological years to maintain the overall symmetry of the tradition. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. === Armenian Eusebius Adjustments === The Armenian Eusebius Chronology: This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or 10 šūši. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] g9hmgayu42gnue0lm3ho1veu8v1fgjx 2803464 2803463 2026-04-08T03:10:17Z CanonicalMormon 2646631 /* Masoretic Adjustments */ 2803464 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the [[Book of Jubilees]]. Under this theory, Isaac's lifespan was increased by six years in the Masoretic, Samaritan, and "Long Chronology" traditions to achieve the mathematical relationship noted above. Conversely, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously. This suggests a deliberate "balancing" of total chronological years to maintain the overall symmetry of the tradition. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. === Armenian Eusebius Adjustments === The Armenian Eusebius Chronology: This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or 10 šūši. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] obmglp0idlm1de5oxkcji9jlsukn5ja 2803465 2803464 2026-04-08T03:14:26Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803465 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the [[wikipedia:Book of Jubilees]]. Under this theory, Isaac's lifespan was increased by six years in the Masoretic, Samaritan, and "Long Chronology" traditions to achieve the mathematical relationship noted above. Conversely, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously. This suggests a deliberate "balancing" of total chronological years to maintain the overall symmetry of the tradition. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. === Armenian Eusebius Adjustments === The Armenian Eusebius Chronology: This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or 10 šūši. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] 4u15icdsxos3m6yr8dcshhwpc79arc7 2803466 2803465 2026-04-08T03:15:02Z CanonicalMormon 2646631 /* Lectio Difficilior Potior */ 2803466 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the [[wikipedia:Book of Jubilees|Book of Jubilees]]. Under this theory, Isaac's lifespan was increased by six years in the Masoretic, Samaritan, and "Long Chronology" traditions to achieve the mathematical relationship noted above. Conversely, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously. This suggests a deliberate "balancing" of total chronological years to maintain the overall symmetry of the tradition. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. === Armenian Eusebius Adjustments === The Armenian Eusebius Chronology: This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or 10 šūši. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] ep6ykryodzu0h302d4go0g3wb3eetxb 2803468 2803466 2026-04-08T03:27:06Z CanonicalMormon 2646631 /* Armenian Eusebius Adjustments */ 2803468 wikitext text/x-wiki {{Original research}} This page evaluates and extends the mathematical insights presented in the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'' by Paul D. While the original article provides a compelling foundation, this analysis identifies areas where the underlying data and mathematical evidence are more robust than initially presented. The following sections aim to clarify these findings and offer a more precise structural framework. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not intended as historical records, but as a complex symbolic mathematical structure designed by ancient authors. Key points include: * '''Artificial Mathematical Design:''' Patriarchal ages and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), the mathematical structures resulted in overlaps where certain patriarchs, such as Methuselah, survived beyond the date of the Flood. = Arichat Yamim = Most of the patriarchs' lifespans in the Hebrew Bible far exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= 3\,\text{šar}\,\,30\,\text{šūši} \\ &= \left(3 \times 60^2 \, \text{years} \right) + \left(30 \times 60^1 \,\text{years} \right) \\ &= 10,800 \, \text{years} + 1,800 \, \text{years} \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">45 šūši<br/>(2700)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">37 šūši<br/>(2220)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 (900) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 (360) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">46 šūši<br/>(2760)</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">32 šūši<br/>(1920)</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 (960) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 (880) | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 (840) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">82 šūši<br/>(4920)</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">40 šūši<br/>(2400)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 (960) | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 (600) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 (420) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">25 šūši<br/>(1500)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 (480) | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 (240) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">17 šūši<br/>(1020)</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 (180) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 (120) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/>(12,600 years) |} == PT2 as the Base Model for Patriarchal Chronologies == The "PT2" chronology serves as the foundational model from which subsequent patriarchal lifespans in various textual traditions were derived. Evidence for this remains visible across nearly all biblical records, as they consistently preserve the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1). * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši'''''. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. The use of rounded Mesopotamian figures in the Armenian Eusebius Chronology suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the Septuagint's divergence indicates a later development—likely in Alexandria—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Josephus<br/>(94 AD) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="3" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2626<br/><small>(2702 - 76)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | — | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | — | 13,200 | 13,551 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> == Comparative Chronology Tables == The following table reconstructs lifespan values across multiple chronological traditions. While most values are derived directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record the lifespans for Levi, Kohath, and Amram. Because these specific values are uniform across other known ''Long Chronology'' traditions, these standard figures have been included here to complete the comparative reconstruction. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="3" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 | 847 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 | 720 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="6" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 777 | 653 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 707 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 723 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 753 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 | rowspan="9" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 | 538 | 535 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | — | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 | 536 | 460 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 | 404 | 567 | colspan="2" | 404 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | colspan="2" | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 | 342 | 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 | colspan="2" | 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 | 198 | 304 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 | 145 | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 174 | colspan="2" | 180 | — | colspan="2" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | rowspan="3" | — | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 | 136 | colspan="2" | 132 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | — | colspan="1" | 13,200 | colspan="1" | 13,551 |} === Masoretic Adjustments === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. notes a potential editorial shift in Lamech's death age: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates a significant tension within the numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 500, and 753) are secondary schematic developments rather than original data. A similar consideration arises with the lifespans of Abraham, Isaac, and Jacob. If one accepts the traditional Masoretic figures, another suspiciously neat pattern emerges: * '''Abraham:''' 175 years (7 × 5<sup>2</sup>) * '''Isaac:''' 180 years (5 × 6<sup>2</sup>) * '''Jacob:''' 147 years (3 × 7<sup>2</sup>) Applying ''Lectio Difficilior'', these figures appear "too structured to be true" and likely represent a late schematic overlay. In the reconstructed prototype chronology (PT2), it is proposed that Isaac's original lifespan was '''174 years'''—a value preserved exclusively in the [[wikipedia:Book of Jubilees|Book of Jubilees]]. Under this theory, Isaac's lifespan was increased by six years in the Masoretic, Samaritan, and "Long Chronology" traditions to achieve the mathematical relationship noted above. Conversely, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously. This suggests a deliberate "balancing" of total chronological years to maintain the overall symmetry of the tradition. === Samaritan Adjustments === Four of the pre-flood patriarchs—Jared, Methuselah, Lamech, and Noah—are credited with exceptionally long lives late in the chronology, which creates a potential overlap with the date of the Deluge. As seen in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. While other traditions do not employ this specific reduction, all chronologies address this potential overlap through various numerical adjustments, as described in later sections. The required reduction to ensure Jared's death coincided with the year of the Flood was '''115 years'''. Interestingly, as noted in a previous section, the Samaritan tradition also reduced the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Isaac's''' lifespan was increased by six years. * '''Amram's''' lifespan was decreased by one year. This net adjustment of 115 years (120 - 6 + 1) suggests a deliberate schematic alignment across the different chronological eras. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's work is dated to This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or 10 šūši. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7<sup>1</sup> = 7 years * '''Jubilee of Years:''' 7<sup>2</sup> = 49 years * '''Week of Jubilees:''' 7<sup>3</sup> = 343 years * '''Jubilee of Jubilees:''' 7<sup>4</sup> = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. ==== Competing Narratives ==== According to the Book of Jubilees 4:33, Shem was the oldest son, born in Noah's 500<sup>th</sup> year, followed by Ham in the 502<sup>nd</sup> year, and Japheth in the 505<sup>th</sup>. This seems to be in contradiction with the Genesis narrative which places Shem as the second son in year 502. ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the harder reading is stronger) suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28%7Cversion=SPE Lamech’s 53rd year]. If we combine that with the 500-year figure for Noah's age at the birth of his sons and the [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB Septuagint figure of 753] for Lamech's death, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing—likely values adjusted during the introduction of the universal flood narrative to create a more "perfect" structure. Applying ''Lectio Difficilior'', we can reasonably set aside the figures 53, 500, and 753 as being "too structured to be true," shifting our focus to less symmetrical values such as '''502''' for Shem's birth. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem is born in year 1207 (reconstructed as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70<sup>th</sup> generation to the 70<sup>th</sup> Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="2" style="background-color:#f9f9f9;" | 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="5" style="background-color:#f9f9f9;" | 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | colspan="9" style="background-color:#e8e8e8;" | 500 .. 502 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="3" | Varied | colspan="1" | 2142 | colspan="1" | 2162 | colspan="1" | 2164 |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1209 | colspan="1" | 1556 | colspan="1" | 1209 | colspan="1" | 2164 | colspan="1" | 2162 | colspan="1" | 2142 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem | colspan="8" style="background-color:#e8e8e8;" | 100 | colspan="1" style="background-color:#e8e8e8;" | 112 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint: <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> While Paul D.’s "Whoops Theory" suggests the LXX editor intended to "fix" the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. === Demetrius the Chronographer === Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. == The Death of Lamech == There are four potential values for Lamech’s original death age: 777 years (MT), 753 years (LXX), 723 years (Eusebius), and 707 years (Josephus). In the comments of the original article, Paul D notes the following regarding Josephus's inconsistency: <blockquote>"Josephus is thought to have used an LXX manuscript similarly corrected in Antiquities 1, but his numbers vary from both MT and LXX in other places, and he has a completely different chronology in Antiquities 8... not to mention some differences between different manuscripts of Josephus."</blockquote> Because Josephus’s figures shift across his own works—suggesting he was revising his estimates—his value of 707 years (which is generally interpreted as a scribal error of 777) lacks the textual weight of the other witnesses. Consequently, we may set aside the Josephan figure to focus on the three primary candidates: 777, 753, and 723. Paul D further suggests: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D accepts 753 as the original age, this conclusion creates a significant tension within his own analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly 12,600 years—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a mere "tweak" in favor of 753 is to overlook the very mathematical architecture that defines the Masoretic tradition. As Paul D acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464 (see Hendel, 1998, pp. 72–73). Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> [[Category:Religion]] p3lsfw9alaulbwoo2494qp3ndg1bpzx WikiJournal of Humanities/Proceedings of the 1st International Scientific and Practical Conference Wikipedia and Wikimedia projects in the focus of scientific research 0 328652 2803374 2803166 2026-04-07T18:32:34Z TMorata 860721 /* Abstracts */ added link 2803374 wikitext text/x-wiki {{under construction}} {{Article info | first1 = Mariana | last1 = Senkiv | orcid1 = 0000-0002-2146-3456 | affiliation1 = Wikimedia Ukraine; Lecturer in the Viacheslav Chornovil Institute of Sustainable Development, Lviv Polytechnic National University: Lviv, Ukraine | submitted= 2025-06-24 | correspondence1 = {{nospam|mariana.senkiv|wikimedia.org.ua}} | journal = WikiJournal of Humanities | w1 = | license = {{CC-BY-SA work}} | abstract = }} [[File:Wikipedia and WikiProjects in the Focus of Research (horizontal).png|thumb|355px|Wikipedia and WikiProjects in the Focus of Research]] ==Foreword== The Proceedings of the 1st International Scientific and Practical Conference “[https://meta.wikimedia.org/wiki/Wikimedia_research_conference_in_Ukraine_2025 Wikipedia and Wiki Projects in the Focus of Research]”, held on November 15, 2025, in Kyiv, Ukraine, in both in-person and online formats, bring together 25 abstracts in Ukrainian dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space. Organized by [https://ua.wikimedia.org/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BC%D0%B5%D0%B4%D1%96%D0%B0_%D0%A3%D0%BA%D1%80%D0%B0%D1%97%D0%BD%D0%B0/en Wikimedia Ukraine], with the support of the Wikimedia Foundation, the conference gathered around 80 participants from Ukraine and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including information quality and reliability, cultural heritage preservation, educational practices, media and information resilience, and technological innovation in wiki projects, while also emphasizing the role of Wikipedia in countering disinformation and preserving knowledge during wartime. The [https://docs.google.com/document/d/18OkSkDhV_KAEij-DF7i1IW1g7UNZu7GX973Ijp9FCjc/edit?tab=t.0 conference program], [https://www.youtube.com/watch?v=lvREKu8eXVI video recordings of presentations], and [[commons:Category:Wikimedia_research_conference_in_Ukraine_2025|photo materials]] are available for further exploration. The Organizing Committee expresses sincere gratitude to all authors, reviewers, partners, and supporters whose contributions ensured the high academic quality of this volume and the success of the conference. ==Abstracts== # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Vasyl Porayko as a case of historical biography in Wikipedia|Vasyl Porayko as a case of historical biography in Wikipedia]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Research_into_the_cultural_heritage_of_Jan_Matejko|Research into the cultural heritage of Jan Matejko in Ukraine through the prism of Wikipedia]] # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Comparative representation of cities in the Ukrainian and Polish Wikipedias|Comparative representation of cities in the Ukrainian and Polish Wikipedias: the cases of Kharkiv and Kraków]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Local_history_wikiprojects_in_Ukraine_as_a_tool_for_digital_encyclopedization_of_local_heritage|Local history wikiprojects in Ukraine as a tool for digital encyclopedization of local heritage]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Wikiprojects_and_cultural_heritage_tourism:_interactions_and_influences|Wikiprojects and cultural heritage tourism: interactions and influences]] # [[WikiJournal of Humanities/Proceedings of the 1st International Scientific and Practical Conference Wikipedia and Wikimedia projects in the focus of scientific research/The unobvious space of war|The unobvious space of war: representations of the past in Wikipedia articles concerning settlements in Lyptsi rural hromada]] # Cooperation between the Wikipedia community and the archives of Ukraine: historical experience and future prospects # Ideological narratives in Russian-language Wikipedia: mediation policy and the representation of Ukrainian history # Wiki Science Competition as a platform for popularization of science (based on materials from 2015–2024) # Information attacks against Wikipedia: analysis of narratives and manipulative tactics on Ukrainian and Russian social media # Wikipedia as a tool for shaping academic integrity in higher education students # Wikipedia and wikiprojects in vocational education pedagogy: tools for media literacy development, opportunities and academic integrity challenges # Wikipedia as a critical reading simulator: from searching for information to creating and editing articles # Using Wikipedia in the education process during language and literature classes # Encyclopedic Wikiresources as a tool for thesauri constructing for learning courses # Wikiprojects as a laboratory of learning # Features of creating the Ukrainian electronic encyclopedia of education using the semantic extension of MediaWiki # The first Ukrainian WikiConferences: at the crossroads between science and education # Geospatial factors of the development of the Crimean Tatar language edition of Wikipedia: analysis of the current situation and opportunities # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Pectoral_from_the_Tovsta_Mohyla|Pectoral from the Tovsta Mohyla on the columns of Wikipedia: verification of the presentation]] # Statistics of music-themed articles in Ukrainian Wikipedia: thematic coverage, quality, and pageviews # Representation of countries of the world and administrative-territorial units in Wikivoyage: analysis of the structure of the Ukrainian-language section # Pedagogy in Ukrainian Wikipedia # Gender gap in biographical articles on the Ukrainian Wikipedia: current state and mitigation strategies # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Demobilization_of_meanings|Demobilization of meanings: how Wikipedia shapes public perception of mobilization processes]] hjnkshtp0vcmk76ka8qp6751vgtqm0c 2803376 2803374 2026-04-07T18:39:08Z TMorata 860721 /* Abstracts */ 2803376 wikitext text/x-wiki {{under construction}} {{Article info | first1 = Mariana | last1 = Senkiv | orcid1 = 0000-0002-2146-3456 | affiliation1 = Wikimedia Ukraine; Lecturer in the Viacheslav Chornovil Institute of Sustainable Development, Lviv Polytechnic National University: Lviv, Ukraine | submitted= 2025-06-24 | correspondence1 = {{nospam|mariana.senkiv|wikimedia.org.ua}} | journal = WikiJournal of Humanities | w1 = | license = {{CC-BY-SA work}} | abstract = }} [[File:Wikipedia and WikiProjects in the Focus of Research (horizontal).png|thumb|355px|Wikipedia and WikiProjects in the Focus of Research]] ==Foreword== The Proceedings of the 1st International Scientific and Practical Conference “[https://meta.wikimedia.org/wiki/Wikimedia_research_conference_in_Ukraine_2025 Wikipedia and Wiki Projects in the Focus of Research]”, held on November 15, 2025, in Kyiv, Ukraine, in both in-person and online formats, bring together 25 abstracts in Ukrainian dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space. Organized by [https://ua.wikimedia.org/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BC%D0%B5%D0%B4%D1%96%D0%B0_%D0%A3%D0%BA%D1%80%D0%B0%D1%97%D0%BD%D0%B0/en Wikimedia Ukraine], with the support of the Wikimedia Foundation, the conference gathered around 80 participants from Ukraine and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including information quality and reliability, cultural heritage preservation, educational practices, media and information resilience, and technological innovation in wiki projects, while also emphasizing the role of Wikipedia in countering disinformation and preserving knowledge during wartime. The [https://docs.google.com/document/d/18OkSkDhV_KAEij-DF7i1IW1g7UNZu7GX973Ijp9FCjc/edit?tab=t.0 conference program], [https://www.youtube.com/watch?v=lvREKu8eXVI video recordings of presentations], and [[commons:Category:Wikimedia_research_conference_in_Ukraine_2025|photo materials]] are available for further exploration. The Organizing Committee expresses sincere gratitude to all authors, reviewers, partners, and supporters whose contributions ensured the high academic quality of this volume and the success of the conference. ==Abstracts== # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Vasyl Porayko as a case of historical biography in Wikipedia|Vasyl Porayko as a case of historical biography in Wikipedia]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Research_into_the_cultural_heritage_of_Jan_Matejko|Research into the cultural heritage of Jan Matejko in Ukraine through the prism of Wikipedia]] # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Comparative representation of cities in the Ukrainian and Polish Wikipedias|Comparative representation of cities in the Ukrainian and Polish Wikipedias: the cases of Kharkiv and Kraków]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Local_history_wikiprojects_in_Ukraine_as_a_tool_for_digital_encyclopedization_of_local_heritage|Local history wikiprojects in Ukraine as a tool for digital encyclopedization of local heritage]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Wikiprojects_and_cultural_heritage_tourism:_interactions_and_influences|Wikiprojects and cultural heritage tourism: interactions and influences]] # [[WikiJournal of Humanities/Proceedings of the 1st International Scientific and Practical Conference Wikipedia and Wikimedia projects in the focus of scientific research/The unobvious space of war|The unobvious space of war: representations of the past in Wikipedia articles concerning settlements in Lyptsi rural hromada]] # Cooperation between the Wikipedia community and the archives of Ukraine: historical experience and future prospects # Ideological narratives in Russian-language Wikipedia: mediation policy and the representation of Ukrainian history # Wiki Science Competition as a platform for popularization of science (based on materials from 2015–2024) # Information attacks against Wikipedia: analysis of narratives and manipulative tactics on Ukrainian and Russian social media # Wikipedia as a tool for shaping academic integrity in higher education students # Wikipedia and wikiprojects in vocational education pedagogy: tools for media literacy development, opportunities and academic integrity challenges # Wikipedia as a critical reading simulator: from searching for information to creating and editing articles # Using Wikipedia in the education process during language and literature classes # Encyclopedic Wikiresources as a tool for thesauri constructing for learning courses # Wikiprojects as a laboratory of learning # Features of creating the Ukrainian electronic encyclopedia of education using the semantic extension of MediaWiki # The first Ukrainian WikiConferences: at the crossroads between science and education # Geospatial factors of the development of the Crimean Tatar language edition of Wikipedia: analysis of the current situation and opportunities # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Pectoral_from_the_Tovsta_Mohyla|Pectoral from the Tovsta Mohyla on the columns of Wikipedia: verification of the presentation]] # Statistics of music-themed articles in Ukrainian Wikipedia: thematic coverage, quality, and pageviews # Representation of countries of the world and administrative-territorial units in Wikivoyage: analysis of the structure of the Ukrainian-language section # Pedagogy in Ukrainian Wikipedia # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Demobilization_of_meanings|Gender gap in biographical articles on the Ukrainian Wikipedia: current state and mitigation strategies]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Demobilization_of_meanings|Demobilization of meanings: how Wikipedia shapes public perception of mobilization processes]] osv03jisqzqrpr6hg2njq0qwkv57ocw 2803377 2803376 2026-04-07T18:40:57Z TMorata 860721 /* Abstracts */ corrected link 2803377 wikitext text/x-wiki {{under construction}} {{Article info | first1 = Mariana | last1 = Senkiv | orcid1 = 0000-0002-2146-3456 | affiliation1 = Wikimedia Ukraine; Lecturer in the Viacheslav Chornovil Institute of Sustainable Development, Lviv Polytechnic National University: Lviv, Ukraine | submitted= 2025-06-24 | correspondence1 = {{nospam|mariana.senkiv|wikimedia.org.ua}} | journal = WikiJournal of Humanities | w1 = | license = {{CC-BY-SA work}} | abstract = }} [[File:Wikipedia and WikiProjects in the Focus of Research (horizontal).png|thumb|355px|Wikipedia and WikiProjects in the Focus of Research]] ==Foreword== The Proceedings of the 1st International Scientific and Practical Conference “[https://meta.wikimedia.org/wiki/Wikimedia_research_conference_in_Ukraine_2025 Wikipedia and Wiki Projects in the Focus of Research]”, held on November 15, 2025, in Kyiv, Ukraine, in both in-person and online formats, bring together 25 abstracts in Ukrainian dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space. Organized by [https://ua.wikimedia.org/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BC%D0%B5%D0%B4%D1%96%D0%B0_%D0%A3%D0%BA%D1%80%D0%B0%D1%97%D0%BD%D0%B0/en Wikimedia Ukraine], with the support of the Wikimedia Foundation, the conference gathered around 80 participants from Ukraine and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including information quality and reliability, cultural heritage preservation, educational practices, media and information resilience, and technological innovation in wiki projects, while also emphasizing the role of Wikipedia in countering disinformation and preserving knowledge during wartime. The [https://docs.google.com/document/d/18OkSkDhV_KAEij-DF7i1IW1g7UNZu7GX973Ijp9FCjc/edit?tab=t.0 conference program], [https://www.youtube.com/watch?v=lvREKu8eXVI video recordings of presentations], and [[commons:Category:Wikimedia_research_conference_in_Ukraine_2025|photo materials]] are available for further exploration. The Organizing Committee expresses sincere gratitude to all authors, reviewers, partners, and supporters whose contributions ensured the high academic quality of this volume and the success of the conference. ==Abstracts== # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Vasyl Porayko as a case of historical biography in Wikipedia|Vasyl Porayko as a case of historical biography in Wikipedia]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Research_into_the_cultural_heritage_of_Jan_Matejko|Research into the cultural heritage of Jan Matejko in Ukraine through the prism of Wikipedia]] # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Comparative representation of cities in the Ukrainian and Polish Wikipedias|Comparative representation of cities in the Ukrainian and Polish Wikipedias: the cases of Kharkiv and Kraków]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Local_history_wikiprojects_in_Ukraine_as_a_tool_for_digital_encyclopedization_of_local_heritage|Local history wikiprojects in Ukraine as a tool for digital encyclopedization of local heritage]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Wikiprojects_and_cultural_heritage_tourism:_interactions_and_influences|Wikiprojects and cultural heritage tourism: interactions and influences]] # [[WikiJournal of Humanities/Proceedings of the 1st International Scientific and Practical Conference Wikipedia and Wikimedia projects in the focus of scientific research/The unobvious space of war|The unobvious space of war: representations of the past in Wikipedia articles concerning settlements in Lyptsi rural hromada]] # Cooperation between the Wikipedia community and the archives of Ukraine: historical experience and future prospects # Ideological narratives in Russian-language Wikipedia: mediation policy and the representation of Ukrainian history # Wiki Science Competition as a platform for popularization of science (based on materials from 2015–2024) # Information attacks against Wikipedia: analysis of narratives and manipulative tactics on Ukrainian and Russian social media # Wikipedia as a tool for shaping academic integrity in higher education students # Wikipedia and wikiprojects in vocational education pedagogy: tools for media literacy development, opportunities and academic integrity challenges # Wikipedia as a critical reading simulator: from searching for information to creating and editing articles # Using Wikipedia in the education process during language and literature classes # Encyclopedic Wikiresources as a tool for thesauri constructing for learning courses # Wikiprojects as a laboratory of learning # Features of creating the Ukrainian electronic encyclopedia of education using the semantic extension of MediaWiki # The first Ukrainian WikiConferences: at the crossroads between science and education # Geospatial factors of the development of the Crimean Tatar language edition of Wikipedia: analysis of the current situation and opportunities # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Pectoral_from_the_Tovsta_Mohyla|Pectoral from the Tovsta Mohyla on the columns of Wikipedia: verification of the presentation]] # Statistics of music-themed articles in Ukrainian Wikipedia: thematic coverage, quality, and pageviews # Representation of countries of the world and administrative-territorial units in Wikivoyage: analysis of the structure of the Ukrainian-language section # Pedagogy in Ukrainian Wikipedia # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Gender_gap_in_biographical_articles_on_the_Ukrainian_Wikipedia|Gender gap in biographical articles on the Ukrainian Wikipedia: current state and mitigation strategies]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Demobilization_of_meanings|Demobilization of meanings: how Wikipedia shapes public perception of mobilization processes]] cj2p1mioif8m5e6g354sa6zv41e3fz2 2803379 2803377 2026-04-07T18:46:21Z TMorata 860721 /* Abstracts */ added link 2803379 wikitext text/x-wiki {{under construction}} {{Article info | first1 = Mariana | last1 = Senkiv | orcid1 = 0000-0002-2146-3456 | affiliation1 = Wikimedia Ukraine; Lecturer in the Viacheslav Chornovil Institute of Sustainable Development, Lviv Polytechnic National University: Lviv, Ukraine | submitted= 2025-06-24 | correspondence1 = {{nospam|mariana.senkiv|wikimedia.org.ua}} | journal = WikiJournal of Humanities | w1 = | license = {{CC-BY-SA work}} | abstract = }} [[File:Wikipedia and WikiProjects in the Focus of Research (horizontal).png|thumb|355px|Wikipedia and WikiProjects in the Focus of Research]] ==Foreword== The Proceedings of the 1st International Scientific and Practical Conference “[https://meta.wikimedia.org/wiki/Wikimedia_research_conference_in_Ukraine_2025 Wikipedia and Wiki Projects in the Focus of Research]”, held on November 15, 2025, in Kyiv, Ukraine, in both in-person and online formats, bring together 25 abstracts in Ukrainian dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space. Organized by [https://ua.wikimedia.org/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BC%D0%B5%D0%B4%D1%96%D0%B0_%D0%A3%D0%BA%D1%80%D0%B0%D1%97%D0%BD%D0%B0/en Wikimedia Ukraine], with the support of the Wikimedia Foundation, the conference gathered around 80 participants from Ukraine and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including information quality and reliability, cultural heritage preservation, educational practices, media and information resilience, and technological innovation in wiki projects, while also emphasizing the role of Wikipedia in countering disinformation and preserving knowledge during wartime. The [https://docs.google.com/document/d/18OkSkDhV_KAEij-DF7i1IW1g7UNZu7GX973Ijp9FCjc/edit?tab=t.0 conference program], [https://www.youtube.com/watch?v=lvREKu8eXVI video recordings of presentations], and [[commons:Category:Wikimedia_research_conference_in_Ukraine_2025|photo materials]] are available for further exploration. The Organizing Committee expresses sincere gratitude to all authors, reviewers, partners, and supporters whose contributions ensured the high academic quality of this volume and the success of the conference. ==Abstracts== # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Vasyl Porayko as a case of historical biography in Wikipedia|Vasyl Porayko as a case of historical biography in Wikipedia]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Research_into_the_cultural_heritage_of_Jan_Matejko|Research into the cultural heritage of Jan Matejko in Ukraine through the prism of Wikipedia]] # [[WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Comparative representation of cities in the Ukrainian and Polish Wikipedias|Comparative representation of cities in the Ukrainian and Polish Wikipedias: the cases of Kharkiv and Kraków]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Local_history_wikiprojects_in_Ukraine_as_a_tool_for_digital_encyclopedization_of_local_heritage|Local history wikiprojects in Ukraine as a tool for digital encyclopedization of local heritage]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Wikiprojects_and_cultural_heritage_tourism:_interactions_and_influences|Wikiprojects and cultural heritage tourism: interactions and influences]] # [[WikiJournal of Humanities/Proceedings of the 1st International Scientific and Practical Conference Wikipedia and Wikimedia projects in the focus of scientific research/The unobvious space of war|The unobvious space of war: representations of the past in Wikipedia articles concerning settlements in Lyptsi rural hromada]] # Cooperation between the Wikipedia community and the archives of Ukraine: historical experience and future prospects # Ideological narratives in Russian-language Wikipedia: mediation policy and the representation of Ukrainian history # Wiki Science Competition as a platform for popularization of science (based on materials from 2015–2024) # Information attacks against Wikipedia: analysis of narratives and manipulative tactics on Ukrainian and Russian social media # Wikipedia as a tool for shaping academic integrity in higher education students # Wikipedia and wikiprojects in vocational education pedagogy: tools for media literacy development, opportunities and academic integrity challenges # Wikipedia as a critical reading simulator: from searching for information to creating and editing articles # Using Wikipedia in the education process during language and literature classes # Encyclopedic Wikiresources as a tool for thesauri constructing for learning courses # Wikiprojects as a laboratory of learning # Features of creating the Ukrainian electronic encyclopedia of education using the semantic extension of MediaWiki # The first Ukrainian WikiConferences: at the crossroads between science and education # Geospatial factors of the development of the Crimean Tatar language edition of Wikipedia: analysis of the current situation and opportunities # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Pectoral_from_the_Tovsta_Mohyla|Pectoral from the Tovsta Mohyla on the columns of Wikipedia: verification of the presentation]] # Statistics of music-themed articles in Ukrainian Wikipedia: thematic coverage, quality, and pageviews # Representation of countries of the world and administrative-territorial units in Wikivoyage: analysis of the structure of the Ukrainian-language section # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Pedagogy_in_Ukrainian_Wikipedia|Pedagogy in Ukrainian Wikipedia]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Gender_gap_in_biographical_articles_on_the_Ukrainian_Wikipedia|Gender gap in biographical articles on the Ukrainian Wikipedia: current state and mitigation strategies]] # [[WikiJournal_of_Humanities/Proceedings/Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research/Demobilization_of_meanings|Demobilization of meanings: how Wikipedia shapes public perception of mobilization processes]] hevjixuzn2h5pq5zlivddoma3ywutwj Complex Analysis/Examples for Power Series 0 328752 2803529 2801659 2026-04-08T09:49:35Z Bert Niehaus 2387134 /* Introduction */ 2803529 wikitext text/x-wiki == Introduction == This page on the topic ''Course:Complex Analysis/Examples of Power Series Developments'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Examples_of_Power_Series_Developments&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Development_into_Power_Series&coursetitle= Wiki2Reveal slides]'''. Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides. The following aspects are considered in detail: * (1) Develop <math>f(z)=\frac{1}{z} </math> [[/Approach/|locally into power series]] and determine the [[w:Radius of convergence|radius of convergence]] via the [[w:Geometric series|geometric series]], which is also essential for the series representation of the [[/Logarithm/|Logarithm]]. * (2) Develop the density of the [[/Standard normal distribution/]] <math>f(x) = \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{x^2}{2}}</math> into a power series in <math>\mathbb{C}</math>. * (3) Develop the [[/Cauchy density/]] <math>f(z)=c\cdot \frac{1}{1+z^2} </math> into a power series in <math>\mathbb{C}</math>. === Basic Approach === The goal of the approach is to generate a representation of the limit <math>\frac{1}{1-q}</math> of a geometric series. Here, <math>q</math> should have the following representation with the development point <math>z_0\in \mathbb{C}</math>: :<math> q:=c\cdot (z-z_o) \, \, \, \, \, \, \, \, \, \, \, \, \frac{1}{1-q} = \sum_{n=0}^\infty \underbrace{c^n \cdot (z-z_0)^n}_{=q^n} {}_{,} </math> With the geometric series, one can directly specify all coefficients <math>c_n:= c^n \in \mathbb{C}</math> without having to calculate the coefficients <math>c_n = \frac{f^{(n)}(z_o)}{n!}</math> individually for the representing [[w:en:Taylor series|Taylor series]]. === Objective === This learning resource on examples of power series developments aims to transfer tools from analysis and series development to power series. In complex analysis<ref>Jänich, K. (2004). Funktionentheorie. Springer Berlin Heidelberg.</ref>, the function <math>f(z)=\frac{1}{z}=z^{-1}</math> or the coefficient <math>(z-z_o)^{-1}</math> in the [[Laurent series]] plays a special role (see [[Course:Complex Analysis/Residue|Residue]]). On <math> \mathbb{C}/\{0\} </math>, <math>f</math> is holomorphic and thus locally developable into power series. In this learning unit, this local development into power series via [[w:Geometric series|geometric series]]<ref>Heuser, H. (2013). Lehrbuch der Analysis: Teil 1. Springer-Verlag.</ref> is discussed. Furthermore, it becomes clear from the representation that the radius of convergence of the power series is the distance between the development point <math> z_o \not= 0</math> and the [[Course:Complex Analysis/Singularities|singularity]] 0. === Geometric Series === The series <math>f(z):= \sum_{n=0}^\infty z^n = \sum_{n=0}^\infty (z-0)^n</math> is a power series with development point 0 and additionally a geometric series with the limit <math> \frac{1}{1-z} </math>. Therefore, the power series <math>f(z)=\sum_{n=0}^\infty z^n=\frac{1}{1-z} </math> represents a power series development of <math> f(z)=\frac{1}{1-z} </math> with development point 0, if <math>|z|<1 </math>. == Tasks for Students == Determine for the above example of the power series development of <math> f(z)=\frac{1}{1-z} </math> with development point <math>z_o = i </math> the first 3 coefficients of the Taylor series development via <math> a_n := \frac{f^{(n)}(i)}{n!} </math>. == Example == In the following example, <math>f(z)=\frac{1}{z}</math> is transformed into a power series development with development point <math>z_o = i</math>. Again, the geometric series is used so that not all coefficients of the Taylor development have to be calculated individually via <math>f^{(n)}(i)</math>. === Transformation into Power Series === :<math> \begin{array}{rcl} \frac{1}{z} & = & -\frac{1}{-z} = - \frac{1}{(i-i)-z} = -\frac{1}{-i-z+i} \\ & = & - \frac{1}{-i-(z-i)} = - \frac{1}{-i\cdot \left(1-\frac{z-i}{-i}\right)} \\ & = & - \underbrace{ \frac{1}{-i}}_{=i} \cdot \frac{1}{ \left(1-\frac{z-i}{-i}\right)} = -i \cdot \sum_{n=0}^{\infty} \left( \frac{z-i}{-i}\right)^n \\ & = & \sum_{n=0}^{\infty} \underbrace{ - i^{n+1} }_{=a_n} \cdot (z-i)^n \end{array} </math> === Radius of Convergence === The series converges for all <math>z\in\mathbb{C}</math> with <math> |z-i| < 1 </math> with <math>|a_n|= 1 </math>. This approach for <math>i</math> is generalized in the following for an arbitrary development point <math>z_o \not=0 </math>. == Task == Generalize the above example for a local Taylor series development of <math>f(z)=\frac{1}{z}</math> around an arbitrary point <math>z_o \in\mathbb{C}</math> with <math>z_o \not=0 </math>. Also, specify the respective radius of convergence of the disk on which the power series converges. == References == <references/> == See Also == * [[Local Development into Power Series]] * [[Residue]] * [[w:Taylor series|Taylor series]] * [[Open Educational Resources]] * [[Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Beispiele für Potenzreihenentwicklungen]]</noinclude> 576lnyba39jbyppk6grn8mmkl28q8gr 2803540 2803529 2026-04-08T10:31:59Z Bert Niehaus 2387134 /* Objective */ 2803540 wikitext text/x-wiki == Introduction == This page on the topic ''Course:Complex Analysis/Examples of Power Series Developments'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Examples_of_Power_Series_Developments&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Development_into_Power_Series&coursetitle= Wiki2Reveal slides]'''. Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides. The following aspects are considered in detail: * (1) Develop <math>f(z)=\frac{1}{z} </math> [[/Approach/|locally into power series]] and determine the [[w:Radius of convergence|radius of convergence]] via the [[w:Geometric series|geometric series]], which is also essential for the series representation of the [[/Logarithm/|Logarithm]]. * (2) Develop the density of the [[/Standard normal distribution/]] <math>f(x) = \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{x^2}{2}}</math> into a power series in <math>\mathbb{C}</math>. * (3) Develop the [[/Cauchy density/]] <math>f(z)=c\cdot \frac{1}{1+z^2} </math> into a power series in <math>\mathbb{C}</math>. === Basic Approach === The goal of the approach is to generate a representation of the limit <math>\frac{1}{1-q}</math> of a geometric series. Here, <math>q</math> should have the following representation with the development point <math>z_0\in \mathbb{C}</math>: :<math> q:=c\cdot (z-z_o) \, \, \, \, \, \, \, \, \, \, \, \, \frac{1}{1-q} = \sum_{n=0}^\infty \underbrace{c^n \cdot (z-z_0)^n}_{=q^n} {}_{,} </math> With the geometric series, one can directly specify all coefficients <math>c_n:= c^n \in \mathbb{C}</math> without having to calculate the coefficients <math>c_n = \frac{f^{(n)}(z_o)}{n!}</math> individually for the representing [[w:en:Taylor series|Taylor series]]. === Objective === This learning resource on examples of power series developments aims to transfer tools from analysis and series development to power series. In complex analysis<ref>Jänich, K. (2004). Funktionentheorie. Springer Berlin Heidelberg.</ref>, the function <math>f(z)=\frac{1}{z}=z^{-1}</math> or the coefficient <math>(z-z_o)^{-1}</math> in the [[Laurent series]] plays a special role (see [[Complex Analysis/Residue|Residue]]). On <math> \mathbb{C}/\{0\} </math>, <math>f</math> is holomorphic and thus locally developable into power series. In this learning unit, this local development into power series via [[w:Geometric series|geometric series]]<ref>Heuser, H. (2013). Lehrbuch der Analysis: Teil 1. Springer-Verlag.</ref> is discussed. Furthermore, it becomes clear from the representation that the radius of convergence of the power series is the distance between the development point <math> z_o \not= 0</math> and the [[Course:Complex Analysis/Singularities|singularity]] 0. === Geometric Series === The series <math>f(z):= \sum_{n=0}^\infty z^n = \sum_{n=0}^\infty (z-0)^n</math> is a power series with development point 0 and additionally a geometric series with the limit <math> \frac{1}{1-z} </math>. Therefore, the power series <math>f(z)=\sum_{n=0}^\infty z^n=\frac{1}{1-z} </math> represents a power series development of <math> f(z)=\frac{1}{1-z} </math> with development point 0, if <math>|z|<1 </math>. == Tasks for Students == Determine for the above example of the power series development of <math> f(z)=\frac{1}{1-z} </math> with development point <math>z_o = i </math> the first 3 coefficients of the Taylor series development via <math> a_n := \frac{f^{(n)}(i)}{n!} </math>. == Example == In the following example, <math>f(z)=\frac{1}{z}</math> is transformed into a power series development with development point <math>z_o = i</math>. Again, the geometric series is used so that not all coefficients of the Taylor development have to be calculated individually via <math>f^{(n)}(i)</math>. === Transformation into Power Series === :<math> \begin{array}{rcl} \frac{1}{z} & = & -\frac{1}{-z} = - \frac{1}{(i-i)-z} = -\frac{1}{-i-z+i} \\ & = & - \frac{1}{-i-(z-i)} = - \frac{1}{-i\cdot \left(1-\frac{z-i}{-i}\right)} \\ & = & - \underbrace{ \frac{1}{-i}}_{=i} \cdot \frac{1}{ \left(1-\frac{z-i}{-i}\right)} = -i \cdot \sum_{n=0}^{\infty} \left( \frac{z-i}{-i}\right)^n \\ & = & \sum_{n=0}^{\infty} \underbrace{ - i^{n+1} }_{=a_n} \cdot (z-i)^n \end{array} </math> === Radius of Convergence === The series converges for all <math>z\in\mathbb{C}</math> with <math> |z-i| < 1 </math> with <math>|a_n|= 1 </math>. This approach for <math>i</math> is generalized in the following for an arbitrary development point <math>z_o \not=0 </math>. == Task == Generalize the above example for a local Taylor series development of <math>f(z)=\frac{1}{z}</math> around an arbitrary point <math>z_o \in\mathbb{C}</math> with <math>z_o \not=0 </math>. Also, specify the respective radius of convergence of the disk on which the power series converges. == References == <references/> == See Also == * [[Local Development into Power Series]] * [[Residue]] * [[w:Taylor series|Taylor series]] * [[Open Educational Resources]] * [[Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Beispiele für Potenzreihenentwicklungen]]</noinclude> mvzsc2zawnmzqda3eanjdwsvukgmuni Complex Analysis/Examples for Power Series/Approach 0 328754 2803538 2801654 2026-04-08T10:25:45Z Bert Niehaus 2387134 2803538 wikitext text/x-wiki == Development into Power Series == Let <math>z_o \not= 0 </math> an arbitrary center for the power series in the complex plane. In this learning resource the representation of the power series is determined with an approach of [[w:Geometric series|geometric series]]. :<math>f(z)=\frac{1}{z}=\sum_{n=0}^\infty a_n\cdot (z-z_o)^n </math> This approach simplifies the calculation of coefficients of the power series in comparison to Taylor coefficients <math>a_n=\tfrac{f^{(n)}}{n!}</math>. The coefficients <math>a_n\in \mathbb{C} </math> are calculated in the following steps. === Step 1 - Geometric Series === The limit of a geometric series is: :<math> \frac{1}{1-q} = \sum_{n=0}^\infty q^n </math> with <math>q \in \mathbb{C} </math> and <math> |q|<1 </math>. This series representation is used in the following steps to obtain the power series by transforming the function term <math>f(z)=\frac{1}{z}</math> into the form <math> \frac{1}{1-q} </math>. === Step 2 - Radius of Convergence === Now, the term <math>f(z)=\frac{1}{z}</math> is transformed into an expression of the form :<math>f(z)=\frac{1}{z}= a\cdot \frac{1}{1-q}</math> where <math>q:=\frac{z-z_o}{b}</math> as a quotient. The complex value <math>q</math> must satisfy the property <math> |q| < 1 </math> resp. <math> |z-z_o| < |b| </math> for the convergence of the geometric series. The [[w:Radius of convergence|radius of convergence]] is <math>r=|b|</math>. === Step 3 - Sign of z === The sign of <math>z</math> must be negative to obtain the structure of the limit for the geometric series <math> \frac{1}{1-q} = \sum_{n=0}^\infty q^n </math> since <math>q</math> is generated as a quotient <math>q=\frac{z-z_o}{b}</math> with <math>b \in \mathbb{C} \setminus \{0\} </math> through transformations. Therefore, <math>\frac{1}{z}</math> is represented in a first step as the following fraction: :<math>f(z)=\frac{1}{z}= -\frac{1}{-z}</math> === Step 4 - Adding Zero === The power series requires the term <math>z-z_o </math> for an arbitrary development point (center) <math>z_o\not= 0</math>. Therefore, zero <math>0 = z_o - z_o </math> is added in the denominator. :<math>f(z)=\frac{1}{z}= -\frac{1}{-z} = -\frac{1}{(z_o-z_o)-z} = -\frac{1}{-z_o-(z-z_o)} </math> === Step 5 - Transformation of Denominator into Limit of Geometric Series === For the transformation of the denominator into the limit of a geometric series <math> \frac{1}{1-q} = \sum_{n=0}^\infty q^n </math> it is necessary to factor out <math>-z_o</math> in the denominator to create a <math>1-q</math> term in the denominator: :<math> \begin{array}{rcl} f(z) & = & -\frac{1}{-z_o-(z-z_o)} = -\frac{1}{(-z_o)\cdot \left( 1-\left( \frac{z-z_o}{-z_o} \right) \right)} \\ & = & \frac{1}{z_o} \cdot \frac{1}{ 1- \underbrace{\frac{z-z_o}{-z_o} }_{=q} } \\ \end{array} </math> === Step 6 - Representation as Geometric Series === The right fraction can be interpreted as the limit <math>\frac{1}{1-q}</math> of a geometric series: :<math>f(z)= \frac{1}{z} = \frac{1}{z_o} \cdot \frac{1}{ 1- \underbrace{\frac{z-z_o}{-z_o} }_{=q} } = \frac{1}{z_o} \cdot \sum_{n=0}^\infty q^n </math> === Step 7 - Representation as Power Series === The geometric series provides the powers <math>(z-z_o)^n</math> for all <math>n\in \mathbb{N}_0</math> and the remaining factor determines the coefficient <math>a_n \in \mathbb{C}</math> of the desired power series: :<math>f(z)= \frac{1}{z} = \frac{1}{z_o} \cdot \frac{1}{ 1- \underbrace{\frac{z-z_o}{-z_o} }_{=q} } = \sum_{n=0}^\infty \underbrace{ \frac{ (-1)^n }{z_o^{n+1}} }_{=a_n} \cdot (z-z_o)^n </math> with <math>|q|:= \left|\frac{z-z_o}{-z_o} \right| < 1 </math>. This yields the [[w:Radius of convergence|radius of convergence]] <math> \left|z-z_o \right| < r:=|-z_o| = |z_o|</math> as the distance from <math> z_o\not= 0 </math> to the [[w:Singularity_(mathematics)#Complex_analysis|singularity]] 0 of <math>f(z)</math>. === Remark - Taylor Series === Alternatively to the above approach, the coefficients <math>a_n</math> can also be calculated via the [[w:Taylor series|Taylor series coefficients]] with <math> a_n= \frac{f^{(n)}(z_o)}{n!} </math>. <span id="Antiderivative_Logarithm"></span> == Branch of the Logarithm - Antiderivative == With the above power series representation of <math>f(z)=\frac{1}{z}</math>, one can develop <math>f</math> around any point <math>z_o\in \mathbb{C}\setminus \{0\}</math> into a power series, where the radius of convergence of the series is <math>r= |z_o| > 0 </math>. Furthermore, one can also specify the power series development of the local antiderivative <math>F_{z_o}</math> of <math>f</math>. :<math>F_{z_o}(z) = \sum_{n=0}^\infty \underbrace{ \frac{ (-1)^n }{z_o^{n+1} \cdot (n+1)}}_{=a_n} \cdot (z-z_o)^{n+1} </math> === Learning Task for Students === On which domain is the [[Complex Analysis/Logarithm|branch of the logarithm]] defined? Create a proof with <math>z_o\in \mathbb{C}\setminus \{0\}</math> and the application of [[w:en:Cauchy–Hadamard_theorem|Cauchy-Hadamard theorem]]! == See Also == * [[Complex Analysis/Examples of Power Series Developments/Cauchy Density|Cauchy Density in the Complex Plane]] * [[Complex Analysis/Logarithm|Logarithm]] <noinclude>[[de:Kurs:Funktionentheorie/Beispiele für Potenzreihenentwicklungen/Vorgehen]]</noinclude> myyf2kbw82ine78trsemoz6gq2hr77k 2803549 2803538 2026-04-08T10:41:23Z Bert Niehaus 2387134 2803549 wikitext text/x-wiki == Development into Power Series == Let <math>z_o \not= 0 </math> an arbitrary center for the power series in the complex plane. In this learning resource the representation of the power series is determined with an approach of [[w:Geometric series|geometric series]]. :<math>f(z)=\frac{1}{z}=\sum_{n=0}^\infty a_n\cdot (z-z_o)^n </math> This approach simplifies the calculation of coefficients of the power series in comparison to Taylor coefficients <math>a_n=\tfrac{f^{(n)}}{n!}</math>. The coefficients <math>a_n\in \mathbb{C} </math> are calculated in the following steps. === Step 1 - Geometric Series === The limit of a geometric series is: :<math> \frac{1}{1-q} = \sum_{n=0}^\infty q^n </math> with <math>q \in \mathbb{C} </math> and <math> |q|<1 </math>. This series representation is used in the following steps to obtain the power series by transforming the function term <math>f(z)=\frac{1}{z}</math> into the form <math> \frac{1}{1-q} </math>. === Step 2 - Radius of Convergence === Now, the term <math>f(z)=\frac{1}{z}</math> is transformed into an expression of the form :<math>f(z)=\frac{1}{z}= a\cdot \frac{1}{1-q}</math> where <math>q:=\frac{z-z_o}{b}</math> as a quotient. The complex value <math>q</math> must satisfy the property <math> |q| < 1 </math> resp. <math> |z-z_o| < |b| </math> for the convergence of the geometric series. The [[w:Radius of convergence|radius of convergence]] is <math>r=|b|</math>. === Step 3 - Sign of z === The sign of <math>z</math> must be negative to obtain the structure of the limit for the geometric series <math> \frac{1}{1-q} = \sum_{n=0}^\infty q^n </math> since <math>q</math> is generated as a quotient <math>q=\frac{z-z_o}{b}</math> with <math>b \in \mathbb{C} \setminus \{0\} </math> through transformations. Therefore, <math>\frac{1}{z}</math> is represented in a first step as the following fraction: :<math>f(z)=\frac{1}{z}= -\frac{1}{-z}</math> === Step 4 - Adding Zero === The power series requires the term <math>z-z_o </math> for an arbitrary development point (center) <math>z_o\not= 0</math>. Therefore, zero <math>0 = z_o - z_o </math> is added in the denominator. :<math>f(z)=\frac{1}{z}= -\frac{1}{-z} = -\frac{1}{(z_o-z_o)-z} = -\frac{1}{-z_o-(z-z_o)} </math> === Step 5 - Transformation of Denominator into Limit of Geometric Series === For the transformation of the denominator into the limit of a geometric series <math> \frac{1}{1-q} = \sum_{n=0}^\infty q^n </math> it is necessary to factor out <math>-z_o</math> in the denominator to create a <math>1-q</math> term in the denominator: :<math> \begin{array}{rcl} f(z) & = & -\frac{1}{-z_o-(z-z_o)} = -\frac{1}{(-z_o)\cdot \left( 1-\left( \frac{z-z_o}{-z_o} \right) \right)} \\ & = & \frac{1}{z_o} \cdot \frac{1}{ 1- \underbrace{\frac{z-z_o}{-z_o} }_{=q} } \\ \end{array} </math> === Step 6 - Representation as Geometric Series === The right fraction can be interpreted as the limit <math>\frac{1}{1-q}</math> of a geometric series: :<math>f(z)= \frac{1}{z} = \frac{1}{z_o} \cdot \frac{1}{ 1- \underbrace{\frac{z-z_o}{-z_o} }_{=q} } = \frac{1}{z_o} \cdot \sum_{n=0}^\infty q^n </math> === Step 7 - Representation as Power Series === The geometric series provides the powers <math>(z-z_o)^n</math> for all <math>n\in \mathbb{N}_0</math> and the remaining factor determines the coefficient <math>a_n \in \mathbb{C}</math> of the desired power series: :<math>f(z)= \frac{1}{z} = \frac{1}{z_o} \cdot \frac{1}{ 1- \underbrace{\frac{z-z_o}{-z_o} }_{=q} } = \sum_{n=0}^\infty \underbrace{ \frac{ (-1)^n }{z_o^{n+1}} }_{=a_n} \cdot (z-z_o)^n </math> with <math>|q|:= \left|\frac{z-z_o}{-z_o} \right| < 1 </math>. This yields the [[w:Radius of convergence|radius of convergence]] <math> \left|z-z_o \right| < r:=|-z_o| = |z_o|</math> as the distance from <math> z_o\not= 0 </math> to the [[w:Singularity_(mathematics)#Complex_analysis|singularity]] 0 of <math>f(z)</math>. === Remark - Taylor Series === Alternatively to the above approach, the coefficients <math>a_n</math> can also be calculated via the [[w:Taylor series|Taylor series coefficients]] with <math> a_n= \frac{f^{(n)}(z_o)}{n!} </math>. <span id="Antiderivative_Logarithm"></span> == Branch of the Logarithm - Antiderivative == With the above power series representation of <math>f(z)=\frac{1}{z}</math>, one can develop <math>f</math> around any point <math>z_o\in \mathbb{C}\setminus \{0\}</math> into a power series, where the radius of convergence of the series is <math>r= |z_o| > 0 </math>. Furthermore, one can also specify the power series development of the local antiderivative <math>F_{z_o}</math> of <math>f</math>. :<math>F_{z_o}(z) = \sum_{n=0}^\infty \underbrace{ \frac{ (-1)^n }{z_o^{n+1} \cdot (n+1)}}_{=a_n} \cdot (z-z_o)^{n+1} </math> === Learning Task for Students === On which domain is the [[Complex Analysis/Logarithm|branch of the logarithm]] defined? Create a proof with <math>z_o\in \mathbb{C}\setminus \{0\}</math> and the application of [[w:en:Cauchy–Hadamard_theorem|Cauchy-Hadamard theorem]]! == See Also == * [[Complex Analysis/Examples of Power Series Developments/Cauchy Density|Cauchy Density in the Complex Plane]] * [[Complex Analysis/Logarithm|Logarithm]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Examples%20for%20Power%20Series/Approach https://en.wikiversity.org/wiki/Complex%20Analysis/Examples%20for%20Power%20Series/Approach] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Examples%20for%20Power%20Series/Approach This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Examples%20for%20Power%20Series/Approach * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Examples%20for%20Power%20Series/Approach&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Approach&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; [[Category:Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Beispiele für Potenzreihenentwicklungen/Vorgehen]]</noinclude> tflqympv9ol2aj7czsmmq9odb2tk8b4 User:Jtneill/Presentations/Open wiki assignments for authentic learning 2 328790 2803451 2803209 2026-04-08T02:23:36Z Jtneill 10242 /* Beyond Wikipedia */ + {{sisterprojects}} 2803451 wikitext text/x-wiki {{title|Open wiki assignments for authentic learning}} <div style="text-align: center"> [[User:Jtneill|James T. Neill]]<br> [[v:University of Canberra|University of Canberra]] [https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]<br> Friday 24 April, 2026 11:00 - 12:00 AEST [https://utsmeet.zoom.us/j/84179400467 Zoom link] <!-- Slides TBA (Google)<br> Video TBA (YouTube; 53:37 mins including Q&A) [https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post] [https://twitter.com/jtneill/status/1768516693553565884 (example) X post] --> </div> ==Overview== {{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}} {{RoundBoxTop|theme=3}} Many student assignments are written for one person, read once, and then never read again. In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio. The session will explore: * What open wiki assignments look like in practice, and where they go wrong * The realities of working in publicly editable spaces (including having work changed or deleted) * Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching. {{RoundBoxBottom}} ==Introduction== A [[w:Wiki|wiki]] is the simplest web page that anyone can edit. Based on this simple idea, wikis have become a cultural phenomenon that seeks to make the sum of all human knowledge freely available to all. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and share the outcomes with the public. Students can use wikis to develop disciplinary knowledge, interact in a dynamic social learning and collaborative editing environment, and to foster generic skills and graduate attributes such as communication skills and being able to make creative use of technology. Staff and students contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. This work is immediately available to the public and can be edited by anyone. Wiki-based assignment formats are flexible and can vary widely depending on subject area, level of study, and targetted skills, but often involve contributing, curating, and improving text or media (images, audio, and video) which can be presented as open educational resources, encyclopedic articles, books, articles, manuals, journals, structured data sets, and so on. Open educational wikis can function as [[w:Content management system|content management systems]] for hosting open teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. While wikis can support the development of open textbooks, they also enable more diverse, collaborative, and participatory forms of knowledge production than institutionally supported textbook platforms such as [[w:PressBooks|PressBooks]]. In the context of [[w:Tertiary education in Australia|Australian higher education]], such platforms are typically staff-controlled, with limited opportunities for student authorship and co-creation. Wikis give students ongoing access to laerning materials beyond their graduation, and staff have access beyond their institutional tenure. Concerns about content curation are resolved by discussion and consenses. Projects can also forked, like open source software, to allow different development directions. ==Wikimedia projects== ===Wikipedia=== The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists. The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. However, I would cast the net wider than Wikipedia because: * Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence. * Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources. For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects. ===Beyond Wikipedia=== Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment. Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development. {{sisterprojects}} In general: * Students contribute discipline-relevant content to the global knowledge commons * Assessment tasks can emphasise creation, curation, synthesis, verification, and/or communication for real audiences * Outputs can include text, media, data, and learning resources * Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages]) Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopaedic entries, and research-informed learning resources. {| class="wikitable" style="margin: 0 auto;" |+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments ! Project ! Purpose ! Example assignments |- | [[b:|Wikibooks]] | New books (e.g., textbooks) | * Contribute to development of an open textbook * Curate and improve existing OER book chapters * Package a series of related articles into a new book |- | [[commons:|Wikimedia Commons]] | Images, audio, and video | * Contribute high-quality educational media * Improve metadata and categorisation * Create educational diagrams and illustrations * Upload field recordings or interviews |- | [[d:Wikidata|Wikidata]] | Structured, linked open data | * Create and curate datasets * Link concepts across Wikimedia projects * Model relationships between entities * Support data-driven research and analysis |- | [[q:|Wikiquote]] | Quotations | * Curate and improve text quotes from primary sources such as political speeches * Create categories for quotes by theme or topic * Add citations and verification to existing quotes |- | [[species:|Wikispecies]] | Taxonomy and species classification | * Curate and improve taxonomic entries for species * Add citations for classification and nomenclature * Contribute information about newly described species * Improve links between species and related Wikimedia projects |- | [[s:Main Page|Wikisource]] | Primary texts and historical documents | * Transcribe and proofread source texts * Annotate and contextualise historical documents * Curate thematic collections of primary sources |- | [[v:|Wikiversity]] | Learning, teaching, and research | * Create open educational resources * Develop teaching materials (e.g., lesson plans, self-assessment quizzes) * Publish student research project summaries * Improve existing learning resources by adding new text and multimedia |- | [[voy:Main Page|Wikivoyage]] | Travel guides and geographic knowledge | * Develop place-based guides (e.g., regions, cities) * Contribute cultural, historical, or environmental information * Integrate fieldwork or experiential learning outputs |- | [[wikt:Main Page|Wiktionary]] | Lexical and linguistic resources | * Create and refine dictionary entries * Analyse word meanings, usage, and etymology * Contribute multilingual translations and examples |- | [[w:|Wikipedia]] | Encyclopedic information | * Contribute to articles related to the class topic where a gap exists * Improve the quality and accuracy of existing articles * Add citations and references to unverified text * Curate and improve a category of articles related to a specific subject area |} ==Open wiki assignments== Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available. Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include: * '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s). * '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators. * '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions. * '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]]. *'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. * '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research. Advantages of open wiki assignments include: * '''Perpetuity''' - ongoing availability of resources * '''Linkability''' - cross-linking of projects and external resources * '''Editability''' - resources can be improved by anyone * '''Discussability''' - each resource has a discussion page * '''Showability''' - resources showcase curator skills and knowledge * '''Transparency''' - resource edit history and can be reviewed * '''Forkability''' - open licence allows development of alternative resources ==Examples== Here are some examples of open wiki assignments: * [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia * [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia * [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA Whilst not assignments per se, these innovative open wiki resources may inspire: * [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries ==Activities== * Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]] * Edit your [[Help:User page|Wikiversity user page]] * Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]] * Brainstorm what you or your students could contribute * Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support ==Bio== [[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''. ==See also== ;Wikimedia Foundation * [[meta:Education|Education]] (Global WMF hub) * [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia) ;Wikipedia * [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta) ;Wikiversity * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article) * [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page) * [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation) [[Category:User:Jtneill/Presentations/Open education]] [[Category:User:Jtneill/Presentations/Wikiversity]] so0gvvmvfu4ls6ay9gu6xu40nkfjghg 2803452 2803451 2026-04-08T02:30:44Z Jtneill 10242 /* Beyond Wikipedia */ Improve paragraph with assistance of ChatGPT: https://chatgpt.com/share/69d5bdc7-78d8-8324-9a77-078d3e66910c 2803452 wikitext text/x-wiki {{title|Open wiki assignments for authentic learning}} <div style="text-align: center"> [[User:Jtneill|James T. Neill]]<br> [[v:University of Canberra|University of Canberra]] [https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]<br> Friday 24 April, 2026 11:00 - 12:00 AEST [https://utsmeet.zoom.us/j/84179400467 Zoom link] <!-- Slides TBA (Google)<br> Video TBA (YouTube; 53:37 mins including Q&A) [https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post] [https://twitter.com/jtneill/status/1768516693553565884 (example) X post] --> </div> ==Overview== {{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}} {{RoundBoxTop|theme=3}} Many student assignments are written for one person, read once, and then never read again. In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio. The session will explore: * What open wiki assignments look like in practice, and where they go wrong * The realities of working in publicly editable spaces (including having work changed or deleted) * Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching. {{RoundBoxBottom}} ==Introduction== A [[w:Wiki|wiki]] is the simplest web page that anyone can edit. Based on this simple idea, wikis have become a cultural phenomenon that seeks to make the sum of all human knowledge freely available to all. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and share the outcomes with the public. Students can use wikis to develop disciplinary knowledge, interact in a dynamic social learning and collaborative editing environment, and to foster generic skills and graduate attributes such as communication skills and being able to make creative use of technology. Staff and students contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. This work is immediately available to the public and can be edited by anyone. Wiki-based assignment formats are flexible and can vary widely depending on subject area, level of study, and targetted skills, but often involve contributing, curating, and improving text or media (images, audio, and video) which can be presented as open educational resources, encyclopedic articles, books, articles, manuals, journals, structured data sets, and so on. Open educational wikis can function as [[w:Content management system|content management systems]] for hosting open teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. While wikis can support the development of open textbooks, they also enable more diverse, collaborative, and participatory forms of knowledge production than institutionally supported textbook platforms such as [[w:PressBooks|PressBooks]]. In the context of [[w:Tertiary education in Australia|Australian higher education]], such platforms are typically staff-controlled, with limited opportunities for student authorship and co-creation. Wikis give students ongoing access to laerning materials beyond their graduation, and staff have access beyond their institutional tenure. Concerns about content curation are resolved by discussion and consenses. Projects can also forked, like open source software, to allow different development directions. ==Wikimedia projects== ===Wikipedia=== The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists. The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. However, I would cast the net wider than Wikipedia because: * Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence. * Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources. For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects. ===Beyond Wikipedia=== Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment. {{sisterprojects}} Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development. An open wiki higher education assignment generally involves: * Students contributing discipline-relevant content to the global knowledge commons via one or more of the Wikimedia Foundation sister projects * Assignment tasks centre on producing and refining useful knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences * Outputs can include text, media, data, and learning resources * Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages]) Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources. {| class="wikitable" style="margin: 0 auto;" |+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments ! Project ! Purpose ! Example assignments |- | [[b:|Wikibooks]] | New books (e.g., textbooks) | * Contribute to development of an open textbook * Curate and improve existing OER book chapters * Package a series of related articles into a new book |- | [[commons:|Wikimedia Commons]] | Images, audio, and video | * Contribute high-quality educational media * Improve metadata and categorisation * Create educational diagrams and illustrations * Upload field recordings or interviews |- | [[d:Wikidata|Wikidata]] | Structured, linked open data | * Create and curate datasets * Link concepts across Wikimedia projects * Model relationships between entities * Support data-driven research and analysis |- | [[q:|Wikiquote]] | Quotations | * Curate and improve text quotes from primary sources such as political speeches * Create categories for quotes by theme or topic * Add citations and verification to existing quotes |- | [[species:|Wikispecies]] | Taxonomy and species classification | * Curate and improve taxonomic entries for species * Add citations for classification and nomenclature * Contribute information about newly described species * Improve links between species and related Wikimedia projects |- | [[s:Main Page|Wikisource]] | Primary texts and historical documents | * Transcribe and proofread source texts * Annotate and contextualise historical documents * Curate thematic collections of primary sources |- | [[v:|Wikiversity]] | Learning, teaching, and research | * Create open educational resources * Develop teaching materials (e.g., lesson plans, self-assessment quizzes) * Publish student research project summaries * Improve existing learning resources by adding new text and multimedia |- | [[voy:Main Page|Wikivoyage]] | Travel guides and geographic knowledge | * Develop place-based guides (e.g., regions, cities) * Contribute cultural, historical, or environmental information * Integrate fieldwork or experiential learning outputs |- | [[wikt:Main Page|Wiktionary]] | Lexical and linguistic resources | * Create and refine dictionary entries * Analyse word meanings, usage, and etymology * Contribute multilingual translations and examples |- | [[w:|Wikipedia]] | Encyclopedic information | * Contribute to articles related to the class topic where a gap exists * Improve the quality and accuracy of existing articles * Add citations and references to unverified text * Curate and improve a category of articles related to a specific subject area |} ==Open wiki assignments== Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available. Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include: * '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s). * '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators. * '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions. * '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]]. *'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. * '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research. Advantages of open wiki assignments include: * '''Perpetuity''' - ongoing availability of resources * '''Linkability''' - cross-linking of projects and external resources * '''Editability''' - resources can be improved by anyone * '''Discussability''' - each resource has a discussion page * '''Showability''' - resources showcase curator skills and knowledge * '''Transparency''' - resource edit history and can be reviewed * '''Forkability''' - open licence allows development of alternative resources ==Examples== Here are some examples of open wiki assignments: * [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia * [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia * [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA Whilst not assignments per se, these innovative open wiki resources may inspire: * [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries ==Activities== * Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]] * Edit your [[Help:User page|Wikiversity user page]] * Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]] * Brainstorm what you or your students could contribute * Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support ==Bio== [[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''. ==See also== ;Wikimedia Foundation * [[meta:Education|Education]] (Global WMF hub) * [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia) ;Wikipedia * [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta) ;Wikiversity * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article) * [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page) * [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation) [[Category:User:Jtneill/Presentations/Open education]] [[Category:User:Jtneill/Presentations/Wikiversity]] mhpccdef42eqno1vdkhr5nuidm7z2bg User talk:Tet-Math3 3 328821 2803550 2802356 2026-04-08T10:48:00Z Tet-Math5 3064239 /* Welcome */ Reply 2803550 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Tet-Math3!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:02, 2 April 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :Hello Jtneill :I have prepared an article in my sandbox at Tet-Math5 :I am seeking to have it examined & hopefully published by Wikiversity BUT it is a potentially dangerous article. It should be examined 1st. [[User:Tet-Math5|Tet-Math5]] ([[User talk:Tet-Math5|discuss]] • [[Special:Contributions/Tet-Math5|contribs]]) 10:48, 8 April 2026 (UTC) 8y3uozt5oft0sshviglo6ku9qmx4hyf Category:Wikiversity books 14 328881 2803557 2803068 2026-04-08T11:24:05Z MarioeMary 3063819 2803557 wikitext text/x-wiki {{Wikipedia|simple:Category:Wikiversity books}} {{Commons|Category:Wikiversity books}} [[Category:Books]] [[Category:Wikiversity]] 16xao84wqx2dbafoxlkdq3hb5g3taas Global Audiology/Input 0 328885 2803372 2803168 2026-04-07T18:20:57Z RadiX 155307 2803372 wikitext text/x-wiki <br /> <div style="border:1px solid #a2a9b1; background:#f8f9fa; padding:15px; border-radius:6px;"> <h3 style="margin-top:0;">🌍 Create a new country page</h3> <p>Type the country name below and click the button to create a standardized page.</p> <inputbox> type=create prefix=Global_Audiology/RecentlyAdded/ preload=Template:Global_Audiology/CountryPreload buttonlabel=Create page placeholder=e.g., Argentina width=40 </inputbox> </div> svrtp0ofl33sg5pbk0zuw27x58w93d8 2803389 2803372 2026-04-07T19:36:30Z RadiX 155307 rv 2803389 wikitext text/x-wiki <div style="border:1px solid #a2a9b1; background:#f8f9fa; padding:15px; border-radius:6px;"> <h3 style="margin-top:0;">🌍 Create a new country page</h3> <p>Type the country name below and click the button to create a standardized page.</p> <inputbox> type=create prefix=Global_Audiology/RecentlyAdded/ preload=Template:Global_Audiology/CountryPreload buttonlabel=Create page placeholder=e.g., Argentina width=40 </inputbox> </div> dwhwyv03bzsyuulh9333dj6mdcz8mxg Template:Global Audiology Authors 10 328888 2803342 2803221 2026-04-07T14:58:34Z RadiX 155307 2803342 wikitext text/x-wiki ubstitua o conteúdo por: <div style=" width:100%; border:1px solid #a2a9b1; border-radius:6px; background:#f8f9fa; padding:10px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:10px; border-bottom:2px solid #36c; padding-bottom:6px; margin-bottom:10px; flex-wrap:wrap; "> <div> [[File:Global Audiology logo.png|70px]] </div> <div style="font-size:1.3em; font-weight:bold;"> Authors </div> </div> <!-- AUTHORS --> <div style="display:flex; flex-wrap:wrap; gap:15px;"> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name1}}}</div> <div style="color:#36c;">{{{role1|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation1|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid1|}}}|<span>🆔 [https://orcid.org/{{{orcid1}}} ORCID]</span>}} {{#if:{{{linkedin1|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin1}}} LinkedIn]</span>}} {{#if:{{{website1|}}}|<span style="margin-left:8px;">🌐 [{{{website1}}} Website]</span>}} {{#if:{{{researchgate1|}}}|<span style="margin-left:8px;">📘 [{{{researchgate1}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name2}}}</div> <div style="color:#36c;">{{{role2|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation2|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid2|}}}|<span>🆔 [https://orcid.org/{{{orcid2}}} ORCID]</span>}} {{#if:{{{linkedin2|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin2}}} LinkedIn]</span>}} {{#if:{{{website2|}}}|<span style="margin-left:8px;">🌐 [{{{website2}}} Website]</span>}} {{#if:{{{researchgate2|}}}|<span style="margin-left:8px;">📘 [{{{researchgate2}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name3}}}</div> <div style="color:#36c;">{{{role3|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation3|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid3|}}}|<span>🆔 [https://orcid.org/{{{orcid3}}} ORCID]</span>}} {{#if:{{{linkedin3|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin3}}} LinkedIn]</span>}} {{#if:{{{website3|}}}|<span style="margin-left:8px;">🌐 [{{{website3}}} Website]</span>}} {{#if:{{{researchgate3|}}}|<span style="margin-left:8px;">📘 [{{{researchgate3}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name4}}}</div> <div style="color:#36c;">{{{role4|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation4|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid4|}}}|<span>🆔 [https://orcid.org/{{{orcid4}}} ORCID]</span>}} {{#if:{{{linkedin4|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin4}}} LinkedIn]</span>}} {{#if:{{{website4|}}}|<span style="margin-left:8px;">🌐 [{{{website4}}} Website]</span>}} {{#if:{{{researchgate4|}}}|<span style="margin-left:8px;">📘 [{{{researchgate4}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name5}}}</div> <div style="color:#36c;">{{{role5|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation5|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid5|}}}|<span>🆔 [https://orcid.org/{{{orcid5}}} ORCID]</span>}} {{#if:{{{linkedin5|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin5}}} LinkedIn]</span>}} {{#if:{{{website5|}}}|<span style="margin-left:8px;">🌐 [{{{website5}}} Website]</span>}} {{#if:{{{researchgate5|}}}|<span style="margin-left:8px;">📘 [{{{researchgate5}}} ResearchGate]</span>}} </div> </div> }} </div> </div> 8x2hu94d1o1479bmbw9s5vxr8twqxok 2803343 2803342 2026-04-07T14:58:46Z RadiX 155307 2803343 wikitext text/x-wiki ubstitua o conteúdo por: <div style=" width:100%; border:1px solid #a2a9b1; border-radius:6px; background:#f8f9fa; padding:10px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:10px; border-bottom:2px solid #36c; padding-bottom:6px; margin-bottom:10px; flex-wrap:wrap; "> <div> [[File:Global Audiology ogo.png|70px]] </div> <div style="font-size:1.3em; font-weight:bold;"> Authors </div> </div> <!-- AUTHORS --> <div style="display:flex; flex-wrap:wrap; gap:15px;"> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name1}}}</div> <div style="color:#36c;">{{{role1|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation1|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid1|}}}|<span>🆔 [https://orcid.org/{{{orcid1}}} ORCID]</span>}} {{#if:{{{linkedin1|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin1}}} LinkedIn]</span>}} {{#if:{{{website1|}}}|<span style="margin-left:8px;">🌐 [{{{website1}}} Website]</span>}} {{#if:{{{researchgate1|}}}|<span style="margin-left:8px;">📘 [{{{researchgate1}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name2}}}</div> <div style="color:#36c;">{{{role2|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation2|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid2|}}}|<span>🆔 [https://orcid.org/{{{orcid2}}} ORCID]</span>}} {{#if:{{{linkedin2|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin2}}} LinkedIn]</span>}} {{#if:{{{website2|}}}|<span style="margin-left:8px;">🌐 [{{{website2}}} Website]</span>}} {{#if:{{{researchgate2|}}}|<span style="margin-left:8px;">📘 [{{{researchgate2}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name3}}}</div> <div style="color:#36c;">{{{role3|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation3|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid3|}}}|<span>🆔 [https://orcid.org/{{{orcid3}}} ORCID]</span>}} {{#if:{{{linkedin3|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin3}}} LinkedIn]</span>}} {{#if:{{{website3|}}}|<span style="margin-left:8px;">🌐 [{{{website3}}} Website]</span>}} {{#if:{{{researchgate3|}}}|<span style="margin-left:8px;">📘 [{{{researchgate3}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name4}}}</div> <div style="color:#36c;">{{{role4|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation4|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid4|}}}|<span>🆔 [https://orcid.org/{{{orcid4}}} ORCID]</span>}} {{#if:{{{linkedin4|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin4}}} LinkedIn]</span>}} {{#if:{{{website4|}}}|<span style="margin-left:8px;">🌐 [{{{website4}}} Website]</span>}} {{#if:{{{researchgate4|}}}|<span style="margin-left:8px;">📘 [{{{researchgate4}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name5}}}</div> <div style="color:#36c;">{{{role5|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation5|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid5|}}}|<span>🆔 [https://orcid.org/{{{orcid5}}} ORCID]</span>}} {{#if:{{{linkedin5|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin5}}} LinkedIn]</span>}} {{#if:{{{website5|}}}|<span style="margin-left:8px;">🌐 [{{{website5}}} Website]</span>}} {{#if:{{{researchgate5|}}}|<span style="margin-left:8px;">📘 [{{{researchgate5}}} ResearchGate]</span>}} </div> </div> }} </div> </div> sfcrt2qxgopg0zxnm8w7iu6mwg947bu 2803344 2803343 2026-04-07T15:00:22Z RadiX 155307 2803344 wikitext text/x-wiki ubstitua o conteúdo por: <div style=" width:100%; border:1px solid #a2a9b1; border-radius:6px; background:#f8f9fa; padding:6px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:10px; border-bottom:2px solid #36c; padding-bottom:6px; margin-bottom:6px; flex-wrap:wrap; "> <div> [[File:Global Audiology ogo.png|70px]] </div> <div style="font-size:1.3em; font-weight:bold;"> Authors </div> </div> <!-- AUTHORS --> <div style="display:flex; flex-wrap:wrap; gap:15px;"> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name1}}}</div> <div style="color:#36c;">{{{role1|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation1|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid1|}}}|<span>🆔 [https://orcid.org/{{{orcid1}}} ORCID]</span>}} {{#if:{{{linkedin1|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin1}}} LinkedIn]</span>}} {{#if:{{{website1|}}}|<span style="margin-left:8px;">🌐 [{{{website1}}} Website]</span>}} {{#if:{{{researchgate1|}}}|<span style="margin-left:8px;">📘 [{{{researchgate1}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name2}}}</div> <div style="color:#36c;">{{{role2|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation2|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid2|}}}|<span>🆔 [https://orcid.org/{{{orcid2}}} ORCID]</span>}} {{#if:{{{linkedin2|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin2}}} LinkedIn]</span>}} {{#if:{{{website2|}}}|<span style="margin-left:8px;">🌐 [{{{website2}}} Website]</span>}} {{#if:{{{researchgate2|}}}|<span style="margin-left:8px;">📘 [{{{researchgate2}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name3}}}</div> <div style="color:#36c;">{{{role3|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation3|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid3|}}}|<span>🆔 [https://orcid.org/{{{orcid3}}} ORCID]</span>}} {{#if:{{{linkedin3|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin3}}} LinkedIn]</span>}} {{#if:{{{website3|}}}|<span style="margin-left:8px;">🌐 [{{{website3}}} Website]</span>}} {{#if:{{{researchgate3|}}}|<span style="margin-left:8px;">📘 [{{{researchgate3}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name4}}}</div> <div style="color:#36c;">{{{role4|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation4|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid4|}}}|<span>🆔 [https://orcid.org/{{{orcid4}}} ORCID]</span>}} {{#if:{{{linkedin4|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin4}}} LinkedIn]</span>}} {{#if:{{{website4|}}}|<span style="margin-left:8px;">🌐 [{{{website4}}} Website]</span>}} {{#if:{{{researchgate4|}}}|<span style="margin-left:8px;">📘 [{{{researchgate4}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name5}}}</div> <div style="color:#36c;">{{{role5|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation5|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid5|}}}|<span>🆔 [https://orcid.org/{{{orcid5}}} ORCID]</span>}} {{#if:{{{linkedin5|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin5}}} LinkedIn]</span>}} {{#if:{{{website5|}}}|<span style="margin-left:8px;">🌐 [{{{website5}}} Website]</span>}} {{#if:{{{researchgate5|}}}|<span style="margin-left:8px;">📘 [{{{researchgate5}}} ResearchGate]</span>}} </div> </div> }} </div> </div> 5irr4efqxxyay7x8beiof755eo8aw5u 2803345 2803344 2026-04-07T15:01:39Z RadiX 155307 2803345 wikitext text/x-wiki <div style=" width:100%; border:1px solid #a2a9b1; border-radius:6px; background:#f8f9fa; padding:6px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:5px; border-bottom:2px solid #36c; padding-bottom:6px; margin-bottom:6px; flex-wrap:wrap; "> <div> [[File:Global Audiology ogo.png|70px]] </div> <div style="font-size:1.3em; font-weight:bold;"> Authors </div> </div> <!-- AUTHORS --> <div style="display:flex; flex-wrap:wrap; gap:15px;"> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name1}}}</div> <div style="color:#36c;">{{{role1|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation1|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid1|}}}|<span>🆔 [https://orcid.org/{{{orcid1}}} ORCID]</span>}} {{#if:{{{linkedin1|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin1}}} LinkedIn]</span>}} {{#if:{{{website1|}}}|<span style="margin-left:8px;">🌐 [{{{website1}}} Website]</span>}} {{#if:{{{researchgate1|}}}|<span style="margin-left:8px;">📘 [{{{researchgate1}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name2}}}</div> <div style="color:#36c;">{{{role2|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation2|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid2|}}}|<span>🆔 [https://orcid.org/{{{orcid2}}} ORCID]</span>}} {{#if:{{{linkedin2|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin2}}} LinkedIn]</span>}} {{#if:{{{website2|}}}|<span style="margin-left:8px;">🌐 [{{{website2}}} Website]</span>}} {{#if:{{{researchgate2|}}}|<span style="margin-left:8px;">📘 [{{{researchgate2}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name3}}}</div> <div style="color:#36c;">{{{role3|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation3|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid3|}}}|<span>🆔 [https://orcid.org/{{{orcid3}}} ORCID]</span>}} {{#if:{{{linkedin3|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin3}}} LinkedIn]</span>}} {{#if:{{{website3|}}}|<span style="margin-left:8px;">🌐 [{{{website3}}} Website]</span>}} {{#if:{{{researchgate3|}}}|<span style="margin-left:8px;">📘 [{{{researchgate3}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name4}}}</div> <div style="color:#36c;">{{{role4|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation4|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid4|}}}|<span>🆔 [https://orcid.org/{{{orcid4}}} ORCID]</span>}} {{#if:{{{linkedin4|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin4}}} LinkedIn]</span>}} {{#if:{{{website4|}}}|<span style="margin-left:8px;">🌐 [{{{website4}}} Website]</span>}} {{#if:{{{researchgate4|}}}|<span style="margin-left:8px;">📘 [{{{researchgate4}}} ResearchGate]</span>}} </div> </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="flex:1 1 350px; border:1px solid #c8ccd1; border-radius:6px; padding:10px; background:white;"> <div style="font-weight:bold;">{{{name5}}}</div> <div style="color:#36c;">{{{role5|}}}</div> <div style="color:#54595d; font-size:0.9em;">{{{affiliation5|}}}</div> <div style="font-size:0.85em; margin-top:6px;"> {{#if:{{{orcid5|}}}|<span>🆔 [https://orcid.org/{{{orcid5}}} ORCID]</span>}} {{#if:{{{linkedin5|}}}|<span style="margin-left:8px;">🔗 [{{{linkedin5}}} LinkedIn]</span>}} {{#if:{{{website5|}}}|<span style="margin-left:8px;">🌐 [{{{website5}}} Website]</span>}} {{#if:{{{researchgate5|}}}|<span style="margin-left:8px;">📘 [{{{researchgate5}}} ResearchGate]</span>}} </div> </div> }} </div> </div> 40d6eiwpyi3l9pjzg430vt4tjyxs0et 2803366 2803345 2026-04-07T18:06:19Z RadiX 155307 reduce white space 2803366 wikitext text/x-wiki <div style=" width:100%; border:1px solid #a2a9b1; border-radius:5px; background:#f8f9fa; padding:8px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:8px; border-bottom:1px solid #36c; padding-bottom:4px; margin-bottom:6px; flex-wrap:wrap; "> <div> [[File:Global Audiology ogo.png|60px]] </div> <div style="font-size:1em; font-weight:bold;"> Contributors to the original text </div> </div> <!-- LIST --> <div style="display:flex; flex-direction:column; gap:4px;"> <!-- AUTHOR TEMPLATE BASE --> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name1}}}</b> {{#if:{{{role1|}}}| • <span style="color:#36c;">{{{role1}}}</span>}} {{#if:{{{affiliation1|}}}| • <span style="color:#54595d;">{{{affiliation1}}}</span>}} {{#if:{{{orcid1|}}}| • 🆔 [https://orcid.org/{{{orcid1}}} ORCID]}} {{#if:{{{linkedin1|}}}| • 🔗 [{{{linkedin1}}} LI]}} {{#if:{{{website1|}}}| • 🌐 [{{{website1}}} Web]}} {{#if:{{{researchgate1|}}}| • 📘 [{{{researchgate1}}} RG]}} </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name2}}}</b> {{#if:{{{role2|}}}| • <span style="color:#36c;">{{{role2}}}</span>}} {{#if:{{{affiliation2|}}}| • <span style="color:#54595d;">{{{affiliation2}}}</span>}} {{#if:{{{orcid2|}}}| • 🆔 [https://orcid.org/{{{orcid2}}} ORCID]}} {{#if:{{{linkedin2|}}}| • 🔗 [{{{linkedin2}}} LI]}} {{#if:{{{website2|}}}| • 🌐 [{{{website2}}} Web]}} {{#if:{{{researchgate2|}}}| • 📘 [{{{researchgate2}}} RG]}} </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name3}}}</b> {{#if:{{{role3|}}}| • <span style="color:#36c;">{{{role3}}}</span>}} {{#if:{{{affiliation3|}}}| • <span style="color:#54595d;">{{{affiliation3}}}</span>}} {{#if:{{{orcid3|}}}| • 🆔 [https://orcid.org/{{{orcid3}}} ORCID]}} {{#if:{{{linkedin3|}}}| • 🔗 [{{{linkedin3}}} LI]}} {{#if:{{{website3|}}}| • 🌐 [{{{website3}}} Web]}} {{#if:{{{researchgate3|}}}| • 📘 [{{{researchgate3}}} RG]}} </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name4}}}</b> {{#if:{{{role4|}}}| • <span style="color:#36c;">{{{role4}}}</span>}} {{#if:{{{affiliation4|}}}| • <span style="color:#54595d;">{{{affiliation4}}}</span>}} {{#if:{{{orcid4|}}}| • 🆔 [https://orcid.org/{{{orcid4}}} ORCID]}} {{#if:{{{linkedin4|}}}| • 🔗 [{{{linkedin4}}} LI]}} {{#if:{{{website4|}}}| • 🌐 [{{{website4}}} Web]}} {{#if:{{{researchgate4|}}}| • 📘 [{{{researchgate4}}} RG]}} </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name5}}}</b> {{#if:{{{role5|}}}| • <span style="color:#36c;">{{{role5}}}</span>}} {{#if:{{{affiliation5|}}}| • <span style="color:#54595d;">{{{affiliation5}}}</span>}} {{#if:{{{orcid5|}}}| • 🆔 [https://orcid.org/{{{orcid5}}} ORCID]}} {{#if:{{{linkedin5|}}}| • 🔗 [{{{linkedin5}}} LI]}} {{#if:{{{website5|}}}| • 🌐 [{{{website5}}} Web]}} {{#if:{{{researchgate5|}}}| • 📘 [{{{researchgate5}}} RG]}} </div> }} <!-- AUTHOR 6 --> {{#if:{{{name6|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name6}}}</b> {{#if:{{{role6|}}}| • <span style="color:#36c;">{{{role6}}}</span>}} {{#if:{{{affiliation6|}}}| • <span style="color:#54595d;">{{{affiliation6}}}</span>}} {{#if:{{{orcid6|}}}| • 🆔 [https://orcid.org/{{{orcid6}}} ORCID]}} {{#if:{{{linkedin6|}}}| • 🔗 [{{{linkedin6}}} LI]}} {{#if:{{{website6|}}}| • 🌐 [{{{website6}}} Web]}} {{#if:{{{researchgate6|}}}| • 📘 [{{{researchgate6}}} RG]}} </div> }} <!-- AUTHOR 7 --> {{#if:{{{name7|}}}| <div style="line-height:1.15; font-size:0.85em;"> <b>{{{name7}}}</b> {{#if:{{{role7|}}}| • <span style="color:#36c;">{{{role7}}}</span>}} {{#if:{{{affiliation7|}}}| • <span style="color:#54595d;">{{{affiliation7}}}</span>}} {{#if:{{{orcid7|}}}| • 🆔 [https://orcid.org/{{{orcid7}}} ORCID]}} {{#if:{{{linkedin7|}}}| • 🔗 [{{{linkedin7}}} LI]}} {{#if:{{{website7|}}}| • 🌐 [{{{website7}}} Web]}} {{#if:{{{researchgate7|}}}| • 📘 [{{{researchgate7}}} RG]}} </div> }} </div> </div> cotgci300k2ro2jg7c1wtcl1l1kekks 2803367 2803366 2026-04-07T18:08:42Z RadiX 155307 upd 2803367 wikitext text/x-wiki <div style=" width:100%; border:1px solid #a2a9b1; border-radius:6px; background:#f8f9fa; padding:6px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:6px; border-bottom:2px solid #36c; padding-bottom:5px; margin-bottom:5px; flex-wrap:wrap; "> <div> [[File:Global Audiology ogo.png|70px]] </div> <div style="font-size:1.15em; font-weight:bold;"> Contributors to the original text </div> </div> <!-- AUTHORS --> <div style="display:flex; flex-wrap:wrap; gap:8px;"> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style=" flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2; "> <div style="font-weight:bold; font-size:0.95em;"> {{{name1}}} </div> <div style="color:#36c; font-size:0.85em;"> {{{role1|}}} </div> <div style="color:#54595d; font-size:0.8em;"> {{{affiliation1|}}} </div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid1|}}}|<span>🆔 [https://orcid.org/{{{orcid1}}} ORCID]</span>}} {{#if:{{{linkedin1|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin1}}} LinkedIn]</span>}} {{#if:{{{website1|}}}|<span style="margin-left:6px;">🌐 [{{{website1}}} Web]</span>}} {{#if:{{{researchgate1|}}}|<span style="margin-left:6px;">📘 [{{{researchgate1}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name2}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role2|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation2|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid2|}}}|<span>🆔 [https://orcid.org/{{{orcid2}}} ORCID]</span>}} {{#if:{{{linkedin2|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin2}}} LinkedIn]</span>}} {{#if:{{{website2|}}}|<span style="margin-left:6px;">🌐 [{{{website2}}} Web]</span>}} {{#if:{{{researchgate2|}}}|<span style="margin-left:6px;">📘 [{{{researchgate2}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name3}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role3|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation3|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid3|}}}|<span>🆔 [https://orcid.org/{{{orcid3}}} ORCID]</span>}} {{#if:{{{linkedin3|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin3}}} LinkedIn]</span>}} {{#if:{{{website3|}}}|<span style="margin-left:6px;">🌐 [{{{website3}}} Web]</span>}} {{#if:{{{researchgate3|}}}|<span style="margin-left:6px;">📘 [{{{researchgate3}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name4}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role4|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation4|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid4|}}}|<span>🆔 [https://orcid.org/{{{orcid4}}} ORCID]</span>}} {{#if:{{{linkedin4|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin4}}} LinkedIn]</span>}} {{#if:{{{website4|}}}|<span style="margin-left:6px;">🌐 [{{{website4}}} Web]</span>}} {{#if:{{{researchgate4|}}}|<span style="margin-left:6px;">📘 [{{{researchgate4}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name5}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role5|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation5|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid5|}}}|<span>🆔 [https://orcid.org/{{{orcid5}}} ORCID]</span>}} {{#if:{{{linkedin5|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin5}}} LinkedIn]</span>}} {{#if:{{{website5|}}}|<span style="margin-left:6px;">🌐 [{{{website5}}} Web]</span>}} {{#if:{{{researchgate5|}}}|<span style="margin-left:6px;">📘 [{{{researchgate5}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 6 --> {{#if:{{{name6|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name6}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role6|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation6|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid6|}}}|<span>🆔 [https://orcid.org/{{{orcid6}}} ORCID]</span>}} {{#if:{{{linkedin6|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin6}}} LinkedIn]</span>}} {{#if:{{{website6|}}}|<span style="margin-left:6px;">🌐 [{{{website6}}} Web]</span>}} {{#if:{{{researchgate6|}}}|<span style="margin-left:6px;">📘 [{{{researchgate6}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 7 --> {{#if:{{{name7|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name7}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role7|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation7|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid7|}}}|<span>🆔 [https://orcid.org/{{{orcid7}}} ORCID]</span>}} {{#if:{{{linkedin7|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin7}}} LinkedIn]</span>}} {{#if:{{{website7|}}}|<span style="margin-left:6px;">🌐 [{{{website7}}} Web]</span>}} {{#if:{{{researchgate7|}}}|<span style="margin-left:6px;">📘 [{{{researchgate7}}} RG]</span>}} </div> </div> }} </div> </div> 6vkbyohrgkibqc0goj0dq0my2f8xqiw 2803369 2803367 2026-04-07T18:17:55Z RadiX 155307 2803369 wikitext text/x-wiki <div style=" width:100%; border:1px solid #a2a9b1; border-radius:6px; background:#f8f9fa; padding:10px; box-sizing:border-box; "> <!-- HEADER --> <div style=" display:flex; align-items:center; gap:10px; border-bottom:2px solid #36c; padding-bottom:6px; margin-bottom:10px; flex-wrap:wrap; "> <div> [[File:Global Audiology ogo.png|70px]] </div> <div style="font-size:1.15em; font-weight:bold;"> Contributors to the original text </div> </div> <!-- AUTHORS --> <div style="display:flex; flex-wrap:wrap; gap:8px;"> <!-- AUTHOR 1 --> {{#if:{{{name1|}}}| <div style=" flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2; "> <div style="font-weight:bold; font-size:0.95em;"> {{{name1}}} </div> <div style="color:#36c; font-size:0.85em;"> {{{role1|}}} </div> <div style="color:#54595d; font-size:0.8em;"> {{{affiliation1|}}} </div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid1|}}}|<span>🆔 [https://orcid.org/{{{orcid1}}} ORCID]</span>}} {{#if:{{{linkedin1|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin1}}} LinkedIn]</span>}} {{#if:{{{website1|}}}|<span style="margin-left:6px;">🌐 [{{{website1}}} Web]</span>}} {{#if:{{{researchgate1|}}}|<span style="margin-left:6px;">📘 [{{{researchgate1}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 2 --> {{#if:{{{name2|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name2}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role2|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation2|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid2|}}}|<span>🆔 [https://orcid.org/{{{orcid2}}} ORCID]</span>}} {{#if:{{{linkedin2|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin2}}} LinkedIn]</span>}} {{#if:{{{website2|}}}|<span style="margin-left:6px;">🌐 [{{{website2}}} Web]</span>}} {{#if:{{{researchgate2|}}}|<span style="margin-left:6px;">📘 [{{{researchgate2}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 3 --> {{#if:{{{name3|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name3}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role3|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation3|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid3|}}}|<span>🆔 [https://orcid.org/{{{orcid3}}} ORCID]</span>}} {{#if:{{{linkedin3|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin3}}} LinkedIn]</span>}} {{#if:{{{website3|}}}|<span style="margin-left:6px;">🌐 [{{{website3}}} Web]</span>}} {{#if:{{{researchgate3|}}}|<span style="margin-left:6px;">📘 [{{{researchgate3}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 4 --> {{#if:{{{name4|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name4}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role4|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation4|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid4|}}}|<span>🆔 [https://orcid.org/{{{orcid4}}} ORCID]</span>}} {{#if:{{{linkedin4|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin4}}} LinkedIn]</span>}} {{#if:{{{website4|}}}|<span style="margin-left:6px;">🌐 [{{{website4}}} Web]</span>}} {{#if:{{{researchgate4|}}}|<span style="margin-left:6px;">📘 [{{{researchgate4}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 5 --> {{#if:{{{name5|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name5}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role5|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation5|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid5|}}}|<span>🆔 [https://orcid.org/{{{orcid5}}} ORCID]</span>}} {{#if:{{{linkedin5|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin5}}} LinkedIn]</span>}} {{#if:{{{website5|}}}|<span style="margin-left:6px;">🌐 [{{{website5}}} Web]</span>}} {{#if:{{{researchgate5|}}}|<span style="margin-left:6px;">📘 [{{{researchgate5}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 6 --> {{#if:{{{name6|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name6}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role6|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation6|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid6|}}}|<span>🆔 [https://orcid.org/{{{orcid6}}} ORCID]</span>}} {{#if:{{{linkedin6|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin6}}} LinkedIn]</span>}} {{#if:{{{website6|}}}|<span style="margin-left:6px;">🌐 [{{{website6}}} Web]</span>}} {{#if:{{{researchgate6|}}}|<span style="margin-left:6px;">📘 [{{{researchgate6}}} RG]</span>}} </div> </div> }} <!-- AUTHOR 7 --> {{#if:{{{name7|}}}| <div style="flex:1 1 300px; border:1px solid #c8ccd1; border-radius:5px; padding:6px 8px; background:white; line-height:1.2;"> <div style="font-weight:bold; font-size:0.95em;">{{{name7}}}</div> <div style="color:#36c; font-size:0.85em;">{{{role7|}}}</div> <div style="color:#54595d; font-size:0.8em;">{{{affiliation7|}}}</div> <div style="font-size:0.75em; margin-top:3px;"> {{#if:{{{orcid7|}}}|<span>🆔 [https://orcid.org/{{{orcid7}}} ORCID]</span>}} {{#if:{{{linkedin7|}}}|<span style="margin-left:6px;">🔗 [{{{linkedin7}}} LinkedIn]</span>}} {{#if:{{{website7|}}}|<span style="margin-left:6px;">🌐 [{{{website7}}} Web]</span>}} {{#if:{{{researchgate7|}}}|<span style="margin-left:6px;">📘 [{{{researchgate7}}} RG]</span>}} </div> </div> }} </div> </div> qrsqmlshg4jy5k8urq32dtj1xv14qjb User:Tet-Math5/sandbox 2 328901 2803537 2803324 2026-04-08T10:25:00Z ~2026-21580-35 3064527 2803537 wikitext text/x-wiki == <math>This\ is\ about\ the\ HIV-AIDS\ virus.</math><math>This\ is\ very\ \ very\ \ very\ \ DANGEROUS. \ Only\ doctors\ should\ attempt\ this\ !!! </math> == === <big><math>But\ what\ if\ there\ IS\ some\ value\ in\ this\ curious\ notion ?</math></big> === .<math>\qquad Several\ decades\ ago\ there\ was\ an\ AIDS\ conference\ here\ in\ London\ Ontario\ (Canada).</math><math>A\ day\ before\ that\ I\ had\ seen\ a\ small\ discussion\ on\ TV\ about\ AIDS.\ \ It\ was\ about\ the\ AIDS\ virus</math><math>being\ very\ sensitive\ to\ room\ temperatures\ that's\ one\ of\ the\ reasons\ why\ people\ can't\ get\ AIDS</math><math>without\ intimte\ contact.</math> <math>\qquad That\ discussion\ left\ me\ with\ the\ impression\ that\ the\ AIDS\ virus\ would\ die\ at\ something\ like</math><math>65^0\ Fahrenheit\ or\ lower\ if\ it\ was\ simply\ left\ on\ a\ table\ top\ or\ in\ an\ exposed\ area\ !!!</math> <math>\qquad Also\ I\ had\ just\ watched\ a\ documentary\ on\ how\ backward\ Russian\ medical\ equipment\ was\ compared\ to\ our</math> <math>high\ tech\ marvels.\ At\ that\ time\ when\ the\ Russian\ doctors\ did\ open\ heart\ surgery\ they\ used\ what\ I\ thought\ was</math><math>an\ astonishing\ process\ that\ enabled\ them\ to\ do\ it.</math> <math>\qquad They\ gave\ their\ patient\ a\ needle\ that\ put\ him\ to\ sleep\ for\ a\ while.\ \ Then\ they\ put\ him\ into\ a\ large\ tub\ of</math><math>very\ cold\ water\ \&\ kept\ adding\ more\ \&\ more\ ice\ cubes\ into\ it\ to\ lower\ the\ patients\ body\ temperature\ down\ to</math><math>something\ like\ 55^0\ Fahrenheit\ or\ so.\quad Close\ to\ \ SUSPENDED\ \ ANIMATION</math> <math>\qquad Then\ they\ took\ his\ body\ out\ of\ the\ ice\ cube\ tub\ \&\ wrapped\ him\ up\ to\ do\ the\ surgery\ on\ what\ looked\ like</math><math>nothing\ more\ than\ a\ kitchen\ table.\ They\ kept\ the\ patients\ body\ temperature\ down\ that\ low\ for\ about\ 40\ minutes</math><math>or\ so.\ Imagine\ doing\ open\ heart\ surgery\ in\ less\ than\ 40\ minutes.</math> <math>\qquad That\ slowed\ the\ patients\ metabolism\ down\ so\ drastically\ that\ the\ patients\ heart\ beat\ very\ mildly\ only\ 2\ or </math><math>maybe\ 3\ times\ per\ minute.\ \ You'd\ think\ only\ a\ yoga\ guy\ could\ do\ something\ like\ that\ \&\ live\ to\ talk\ about\ it. </math> <math>The\ point\ I'm\ getting\ to\ is\ this: </math><math>\qquad Would\ lowering\ the\ body\ temperature\ of\ an\ AIDS\ victim\ below\ the\ kill\ point\ of\ AIDS\ cure\ AIDS\ ? </math>. <math>In\ the\ 1990's\ this\ letter\ was\ sent\ to\ the\ HIV-AIDS\ conference\ that\ was\ held\ in\ London.\ But\ to\ no\ effect.</math> == <big><math>Suspended\ animation.</math></big> == <big>https://litfl.com/suspended-animation/</big> elxl4l6konmytrh5bo2b4i6bpddaucq File:VLSI.Arith.2A.CLA.20260407.pdf 6 328902 2803328 2026-04-07T13:38:31Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260407 - 20260406) |Source={{own|Young1lim}} |Date=2026-04-07 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2803328 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260407 - 20260406) |Source={{own|Young1lim}} |Date=2026-04-07 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} gauy9r9jdufixy73p8qnrbfeo2bj31b File:C04.SA0.PtrOperator.1A.20260407.pdf 6 328903 2803333 2026-04-07T13:54:56Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260407 - 20260406) |Source={{own|Young1lim}} |Date=2026-04-07 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2803333 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260407 - 20260406) |Source={{own|Young1lim}} |Date=2026-04-07 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 8d2tdz7z67lkzdsbrs7cvkqyl4dcdno File:Laurent.5.Permutation.6C.20260406.pdf 6 328904 2803335 2026-04-07T14:15:04Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260406 - 20260404) |Source={{own|Young1lim}} |Date=2026-04-06 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2803335 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260406 - 20260404) |Source={{own|Young1lim}} |Date=2026-04-06 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} e1gjljk15cu1ryrpa62545ra27duo3h WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Demobilization of meanings 0 328915 2803373 2026-04-07T18:30:41Z TMorata 860721 uploaded content 2803373 wikitext text/x-wiki {{Article info | first1 = Roman | last1 = Khardel | orcid1 = 0000-0002-2243-9722 | affiliation1 = Hetman Petro Sahaidachnyi National Army Academy, Lviv, UA | correspondence1 = {{nospam|spantamano2|gmail.com}} | w1 = | journal = WikiJournal of Humanities | license = | abstract = | submitted = 2025-06-24 }} '''Abstract''' The article examines how the Ukrainian-language Wikipedia represents the concepts of “mobilization” and “demobilization.” Legislative and encyclopedic definitions are compared with the actual content of Wikipedia entries. The study reveals a tendency to reduce the complex state process of mobilization to its partial element – military conscription. It also shows that demobilization is interpreted mainly as individual discharge from service rather than as the systemic winding down of state mobilization measures. Such interpretive simplifications shape public understanding of mobilization as both a wartime and governance phenomenon. ''Keywords:'' mobilization, demobilization, Wikipedia, information warfare, media discourse, public perception. У тезах досліджено відображення понять «мобілізація» та «демобілізація» в україномовній Вікіпедії. Порівняно законодавчі та енциклопедичні визначення з реальним змістом вікістатей. Виявлено тенденцію редукції складного державного процесу мобілізації до його часткового елементу – військового призову. Показано, що демобілізація у Вікіпедії трактується переважно як індивідуальне звільнення зі служби, а не як системне згортання державних мобілізаційних заходів. Доведено, що такі інтерпретаційні спрощення впливають на суспільне сприйняття мобілізації як феномена війни та державного управління. ''Ключові слова:'' мобілізація, демобілізація, Вікіпедія, інформаційна війна, медіадискурс, суспільне уявлення. 6slcj1vd1wy6f7uxvc0ns2i58jy6ff1 WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Gender gap in biographical articles on the Ukrainian Wikipedia 0 328916 2803375 2026-04-07T18:37:20Z TMorata 860721 uploaded content 2803375 wikitext text/x-wiki {{Article info | first1 = Anton | last1 = Protsiuk | orcid1 = 0009-0007-5929-6722 | affiliation1 = National Academy of Sciences of Ukraine, Kyiv, UA | correspondence1 = {{nospam|anton.protsiuk|wikimedia.org.ua}} | w1 = | journal = WikiJournal of Humanities | license = | abstract = | submitted = 2025-06-24 }} '''Abstract''' The study examines the current level of gender gap in biographical articles of the Ukrainian-language edition of Wikipedia and its historical dynamics, and analyzes thematic campaigns run by Wikimedia Ukraine as a key form of organized efforts to address the issue. The findings indicate that the gender gap in Ukrainian Wikipedia has been gradually narrowing, and that campaigns by Wikimedia Ukraine are an important – though by no means the only – factor in this process. The analysis underscores the need for further, more detailed research on the topic, including the effectiveness of thematic campaigns in reducing the gender gap. ''Key words:'' Wikipedia, Wikidata, gender gap, thematic campaigns on Wikipedia Дослідження аналізує поточний рівень гендерного дисбалансу у біографічних статтях україномовного розділу Вікіпедії та його історичну динаміку, а також вивчає тематичні кампанії громадської організації «Вікімедіа Україна» як ключовий спосіб організованої роботи із вирішення проблеми. Результати показують, що гендерний дисбаланс в українській Вікіпедії поступово скорочується, і кампанії ГО «Вікімедіа Україна» є важливим, але далеко не єдиним фактором у цьому процесі. Аналіз показує потребу в подальшому детальнішому вивченні теми, зокрема ефективності тематичних кампаній у скороченні гендерного дисбалансу. ''Ключові слова:'' Вікіпедія, Вікідані, гендерний дисбаланс, тематичні кампанії у Вікіпедії. sxyxas743pzypmv13gq5gww02oc7ztm WikiJournal of Humanities/Proceedings/Wikipedia and Wikimedia projects in the focus of scientific research/Pedagogy in Ukrainian Wikipedia 0 328917 2803378 2026-04-07T18:45:19Z TMorata 860721 uploaded content 2803378 wikitext text/x-wiki {{Article info | first1 = Oleksandr | last1 = Zheliba | orcid1 = 0000-0002-7711-874X | affiliation1 = Nizhyn Gogol State University, Nizhyn, UA | correspondence1 = {{nospam|geliba|ukr.net}} | w1 = | journal = WikiJournal of Humanities | license = | abstract = | submitted = 2025-06-24 }} '''Abstract''' The article provides a comprehensive analysis of how pedagogy is represented in the Ukrainian Wikipedia: quantitative and qualitative characteristics of articles, comparison with the English-language section, the impact of contests and campaigns. Using PetScan and content analysis, the study highlights the English segment’s advantages and the Ukrainian section’s national focus. Recommendations include integrating Wikipedia editing into teacher education, strengthening scholar–community cooperation. ''Keywords:'' pedagogy; Wikipedia; digital resources; content quality; media literacy. Автор комплексно аналізує представлення педагогічної тематики в Українській Вікіпедії: кількісні й якісні показники статей, порівняння з англомовним розділом, роль конкурсів і тематичних кампаній. На основі PetScan і контент-аналізу виявлено переваги англомовного контенту та національну спрямованість українського розділу. Надано рекомендації щодо інтеграції редагування Вікіпедії в освіту, посилення співпраці науковців і спільноти. ''Ключові слова:'' педагогіка; Вікіпедія; цифрові ресурси; якість контенту; медіаграмотність. t24231crfoghe6ekgw1d8zbawv70hjk File:THE EXISTENCE OF HILBERT-PÓLYA OPERATOR.pdf 6 328918 2803405 2026-04-07T20:35:16Z Amal Ladjeroud 3062989 {{Information |Description=A complete proof to the Hilbert-Pólya conjecture |Source= |Date=1 November 2025 |Author=Amal Ladjeroud |Permission= }} 2803405 wikitext text/x-wiki == Summary == {{Information |Description=A complete proof to the Hilbert-Pólya conjecture |Source= |Date=1 November 2025 |Author=Amal Ladjeroud |Permission= }} == Licensing == {{self|GFDL|cc-by-4.0}} 5j9flv32klrdvo98tl2c0ch5q3aa2vv 2803407 2803405 2026-04-07T20:38:30Z Amal Ladjeroud 3062989 /* Licensing */ CC-BY 4.0 2803407 wikitext text/x-wiki == Summary == {{Information |Description=A complete proof to the Hilbert-Pólya conjecture |Source=Amal Ladjeroud |Date=1 November 2025 |Author=Amal Ladjeroud |Permission= }} == Licensing == {{self|GFDL|cc-by-4.0}} tuyy8ijp7q2jfxhigbsy7ojr2k0oqq9 User:Amal Ladjeroud 2 328919 2803413 2026-04-07T20:47:34Z Amal Ladjeroud 3062989 Created blank page 2803413 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 Digital Media Concepts/Use of CGI in Motion Picture Film How to Train Your Dragon 0 328920 2803444 2026-04-07T23:32:44Z Jaylor22 3061875 Created page with "== Introduction == How to train your dragon was originally an animated trilogy with their first film being released on March 21, 2010. It became an immense critical and commercial success. Bringing in roughly $495 million worldwide. That was impressive considering they had a $165 million budget. It’s no surprise why they decided to make 2 more moves and a spin off series. Although their biggest successes had to be the 2025 live action remake. It brought in roughly $636..." 2803444 wikitext text/x-wiki == Introduction == How to train your dragon was originally an animated trilogy with their first film being released on March 21, 2010. It became an immense critical and commercial success. Bringing in roughly $495 million worldwide. That was impressive considering they had a $165 million budget. It’s no surprise why they decided to make 2 more moves and a spin off series. Although their biggest successes had to be the 2025 live action remake. It brought in roughly $636 million, and it became the second-highest-grossing live-action/animated hybrid film of all time. Though a discussion everyone seemed to have was, how realistic the created were able to make them. == The CGI Process == The dragons we loved as kids were able to come to life due to the use of CGI and some fancy puppets. Making animated dragons seems as if they came to life is not an easy job. That is why “''Framestore visual effects and computer animation studio was chosen to be the primary vendor that would actualize Toothless and the training dragons against the human actors” (-NBC)''<ref>{{Cite web|url=https://www.nbc.com/nbc-insider/how-to-train-your-dragon-live-action-movie-weaved-cgi-with-practical-effects|title=How to Train Your Dragon’s Director Details Updating The Dragons for Live-Action|date=2025-06-06|website=NBC|language=en-US|access-date=2026-04-07}}</ref>''.'' That is the same company that helped Harry Potter bring their creatures to life. == The use of Puppets == Although CGI wasn't the only component to the success of the dragons. It was clear to the director that only using CGI would not translate well the audience. So their solution was to create fully functional dragon heads for more up close scenes. For scenes like the one where Hiccup needs to pet Toothless, they would use that head so the animators had an easier time making their dragons seem more alive. Something they also took into consideration was that the production team wanted the audience to see these creatures and find some similarities to their pets. There is where that puppet head also helps == Sources == https://www.nbc.com/nbc-insider/how-to-train-your-dragon-live-action-movie-weaved-cgi-with-practical-effects https://www.cnet.com/tech/services-and-software/how-to-train-your-dragon-has-tempered-my-disdain-for-live-action-remakes/ https://thecollegiatelive.com/2025/06/from-animation-to-awe-the-live-action-how-to-train-your-dragon-delivers/ <nowiki>https://youtu.be/S0yxE65n_XY?si=9_6GoXz6Mc7WqcOT</nowiki> <nowiki>https://youtu.be/uf81rB2Adm8?si=Qoi4E4jjzzdARCsG</nowiki> eu8ml99cxcz7a6hqvcy2xo8gvutmr4j User talk:~2026-21347-26 3 328921 2803471 2026-04-08T04:16:15Z PieWriter 3039865 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2803471 wikitext text/x-wiki == 2026-04-08 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.<!-- Glow-vandalism1 @ 1775621774838s --><nowiki></nowiki> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:16, 8 April 2026 (UTC) hfsrolt8lf4lgtz7y86v5mvu2hkivwy Portal:Bikol 102 328922 2803473 2026-04-08T05:53:19Z CarlessParking 3064444 Created page with "Welcome to the Bikol Department at Wikiversity. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speak..." 2803473 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ifh8qo5lp08wf3fhg8u4m73225iltc4 2803492 2803473 2026-04-08T06:57:05Z CarlessParking 3064444 2803492 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. == Division News == * '''08 April 2026 ''' - Department founded! pisblnuhvu0ir137jxufqb2aridu6jv 2803493 2803492 2026-04-08T06:59:19Z CarlessParking 3064444 2803493 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]].. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. == Division News == * '''08 April 2026 ''' - Department founded! jy536opy2zomg9ydz0unvk0dt2wj353 2803494 2803493 2026-04-08T06:59:44Z CarlessParking 3064444 2803494 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. == Division News == * '''08 April 2026 ''' - Department founded! 8yigny0t7n0r00owcbl7cr5x6hh85m2 2803496 2803494 2026-04-08T07:10:13Z CarlessParking 3064444 2803496 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== == Division News == * '''08 April 2026 ''' - Department founded! 3urn8ipbz8h72k6g0vquxwm8lbhjrci 2803499 2803496 2026-04-08T07:22:43Z CarlessParking 3064444 /* Courses */ 2803499 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will how to greet and introduce yourself in Bikol. == Division News == * '''08 April 2026 ''' - Department founded! mnsjsuryochdypgw6gnio8ydqz9ccjw 2803500 2803499 2026-04-08T07:22:56Z CarlessParking 3064444 /* Courses */ 2803500 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. == Division News == * '''08 April 2026 ''' - Department founded! 67zhj7qetmerxmnor3vrwd3pfdsq7bz 2803501 2803500 2026-04-08T07:23:16Z CarlessParking 3064444 2803501 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! 02xarx2c08ebp54gyhk9b6hn1qgajr1 2803502 2803501 2026-04-08T07:34:54Z CarlessParking 3064444 /* Courses */ 2803502 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ngt80qlaszb45wtvuxd4xgtvgrwzfl8 2803507 2803502 2026-04-08T08:03:16Z CarlessParking 3064444 /* Courses */ 2803507 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | {{wikt|Index:Central Bikol|Word index}} at the English Wiktionary |}</div> == Division News == * '''April 8, 2026 ''' - Department founded! 11w7h0ih5hotrq7tjxwhjdmai5z78zm 2803508 2803507 2026-04-08T08:06:28Z CarlessParking 3064444 /* Courses */ 2803508 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | {{wikv|Bikol phrasebook|Bikol}} a phrasebook at Wikivoyage |}</div> == Division News == * '''April 8, 2026 ''' - Department founded! so73jhjjvt4oui3e6wzwreu065yg9pl 2803511 2803508 2026-04-08T08:14:05Z CarlessParking 3064444 /* Courses */ 2803511 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | [[:b:Wikivoyage|Bikol phrasebook]] at Wikivoyage |}</div> == Division News == * '''April 8, 2026 ''' - Department founded! cglb12umu1pbhqbfj4npzgr9xhupplc 2803512 2803511 2026-04-08T08:14:33Z CarlessParking 3064444 2803512 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ngt80qlaszb45wtvuxd4xgtvgrwzfl8 2803513 2803512 2026-04-08T08:14:56Z CarlessParking 3064444 /* Division News */ 2803513 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ==External Links== <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | [[:b:Wikivoyage|Bikol phrasebook]] at Wikivoyage |}</div> a0bs4x6w18woseim3b9jwn1f2trg0x0 2803514 2803513 2026-04-08T08:15:26Z CarlessParking 3064444 2803514 wikitext text/x-wiki Welcome to the Bikol Department at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ==External Links== <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | [[:b:Wikivoyage|Bikol phrasebook]] at Wikivoyage |}</div> qcip7mt1cp7pc22f4xc24bur4vs4o9j 2803530 2803514 2026-04-08T09:51:55Z CarlessParking 3064444 2803530 wikitext text/x-wiki Welcome to the '''Bikol Department''' at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ==External Links== <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | [[:b:Wikivoyage|Bikol phrasebook]] at Wikivoyage |}</div> 5fgjy5xsg58yvkdomtuef8psnfmx1j7 2803536 2803530 2026-04-08T10:06:22Z CarlessParking 3064444 /* Courses */ 2803536 wikitext text/x-wiki Welcome to the '''Bikol Department''' at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. ==Introduction== Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca. ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Grammar|Grammar]] - this lesson will answer your curiosity on how words are being joined together in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ==External Links== <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | [[:b:Wikivoyage|Bikol phrasebook]] at Wikivoyage |}</div> onalyvvg8rept50ls23ss46kkiv0ldr 2803558 2803536 2026-04-08T11:42:27Z CarlessParking 3064444 /* Introduction */ 2803558 wikitext text/x-wiki Welcome to the '''Bikol Department''' at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]]. <div class="noprint"><!-- class="noprint" makes this box disappear when printing --> {| cellpadding="1" align="center" style="background-color: #fffff0; border: 1px solid #FFBF00; padding: 0.5em; font-size: big; text-align: center;" | Bikol is an Austronesian language used in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca.</div> ==Courses== * [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol. * [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Grammar|Grammar]] - this lesson will answer your curiosity on how words are being joined together in Bikol. == Division News == * '''April 8, 2026 ''' - Department founded! ==External Links== <div style="float: right; margin: 2px;"> {| class="wikitable" border="1" |- |{{center|'''Additional Wikimedia resources'''}} |- ! style="background: #lime" colspan="6" | [[:b:Bikol|Textbook]] at Wikibooks |- ! style="background: #lime" colspan="6" | {{w|Central Bikol|Article}} at Wikipedia |- ! style="background: #lime" colspan="6" | [[:b:Wikivoyage|Bikol phrasebook]] at Wikivoyage |}</div> 2h52z9z65j3hos0s9wkn9j8x75qkzgh Southeast Asian Languages/Philippine Languages/Bikol 0 328923 2803476 2026-04-08T05:59:15Z CarlessParking 3064444 Created page with "__NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big..." 2803476 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Mabuhay!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] hjx8cpwv3op44xowvb7y4dfb854av0c 2803477 2803476 2026-04-08T06:06:03Z CarlessParking 3064444 2803477 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Mabuhay!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. l5obwv8s4a1kosjmikhkvaigjfn0ajs 2803478 2803477 2026-04-08T06:08:46Z CarlessParking 3064444 2803478 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Mabuhay!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] ==Bikol 1== [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]], a foundational course provided by the [[Portal:Southeast Asian languages|Division of Southeast Asian Languages]], serves as an entry point for students seeking to acquaint themselves with the rudimentary aspects of the language, encompassing uncomplicated vocabulary acquisition, the mastering of fundamental phrases, and the exploration of elementary grammatical structures. Throughout this course, participants embark on an immersive linguistic journey, progressively building their proficiency in Filipino through engaging lessons, interactive exercises, and practical conversational practice that empowers them to communicate effectively in real-world scenarios. | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. b39yf1ycl98blgtdggwnukduac6zjgq 2803479 2803478 2026-04-08T06:11:03Z CarlessParking 3064444 2803479 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Mabuhay!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] ==Bikol 1== [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]], a foundational course provided by the [[Portal:Southeast Asian languages|Division of Southeast Asian Languages]], serves as an entry point for students seeking to acquaint themselves with the rudimentary aspects of the language, encompassing uncomplicated vocabulary acquisition, the mastering of fundamental phrases, and the exploration of elementary grammatical structures. Throughout this course, participants embark on an immersive linguistic journey, progressively building their proficiency in Filipino through engaging lessons, interactive exercises, and practical conversational practice that empowers them to communicate effectively in real-world scenarios. | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. |- | style="width: 60%; background-color: Cornsilk; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | [[Image:Nuvola apps edu miscellaneous.svg|right|96px]] ==Lessons== * Lesson #1<nowiki>: </nowiki>'''[[/Lesson:Vocabulary|Vocabulary]]''' |} {{Category:Bikol}} hq6zjis85fembcfupa1dxa9hbnivgpu 2803480 2803479 2026-04-08T06:11:56Z CarlessParking 3064444 2803480 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Mabuhay!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] ==Bikol 1== [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]], a foundational course provided by the [[Portal:Southeast Asian languages|Division of Southeast Asian Languages]], serves as an entry point for students seeking to acquaint themselves with the rudimentary aspects of the language, encompassing uncomplicated vocabulary acquisition, the mastering of fundamental phrases, and the exploration of elementary grammatical structures. Throughout this course, participants embark on an immersive linguistic journey, progressively building their proficiency in Filipino through engaging lessons, interactive exercises, and practical conversational practice that empowers them to communicate effectively in real-world scenarios. | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. |- | style="width: 60%; background-color: Cornsilk; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | [[Image:Nuvola apps edu miscellaneous.svg|right|96px]] ==Lessons== * Lesson #1<nowiki>: </nowiki>'''[[/Lesson:Vocabulary|Vocabulary]]''' |} [[Category:Bikol]] e6aml9uvlpv5jht7el3njczp6w1r37i 2803503 2803480 2026-04-08T07:39:49Z CarlessParking 3064444 2803503 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Welcome!</big>{{center bottom}} {{center top}}<big>Maugmang Pag-abot!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] ==Bikol 1== [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]], a foundational course provided by the [[Portal:Southeast Asian languages|Division of Southeast Asian Languages]], serves as an entry point for students seeking to acquaint themselves with the rudimentary aspects of the language, encompassing uncomplicated vocabulary acquisition, the mastering of fundamental phrases, and the exploration of elementary grammatical structures. Throughout this course, participants embark on an immersive linguistic journey, progressively building their proficiency in Filipino through engaging lessons, interactive exercises, and practical conversational practice that empowers them to communicate effectively in real-world scenarios. | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. |- | style="width: 60%; background-color: Cornsilk; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | [[Image:Nuvola apps edu miscellaneous.svg|right|96px]] ==Lessons== * Lesson #1<nowiki>: </nowiki>'''[[/Lesson:Vocabulary|Vocabulary]]''' |} [[Category:Bikol]] 4c1noqpyroqd5xl719p78faghxzi579 2803532 2803503 2026-04-08T09:58:08Z CarlessParking 3064444 2803532 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Welcome!</big>{{center bottom}} {{center top}}<big>Maugmang Pag-abot!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] ==Bikol 1== [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]], a foundational course provided by the [[Portal:Southeast Asian languages|Division of Southeast Asian Languages]], serves as an entry point for students seeking to acquaint themselves with the rudimentary aspects of the language, encompassing uncomplicated vocabulary acquisition, the mastering of fundamental phrases, and the exploration of elementary grammatical structures. Throughout this course, participants embark on an immersive linguistic journey, progressively building their proficiency in Filipino through engaging lessons, interactive exercises, and practical conversational practice that empowers them to communicate effectively in real-world scenarios. | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. |- | style="width: 60%; background-color: Cornsilk; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | [[Image:Nuvola apps edu miscellaneous.svg|right|96px]] ==Lessons== * Lesson #1<nowiki>: </nowiki>'''[[/Lesson:Vocabulary|Vocabulary]]''' * Lesson #2<nowiki>:</nowiki>'''[[/Lesson:Grammar|Grammar]]''' |} [[Category:Bikol]] m9ai03ru0bmbjhpdnf1dyguq1pq5n3z 2803535 2803532 2026-04-08T10:03:46Z CarlessParking 3064444 2803535 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {| cellpadding="10" cellspacing="5" style="width: 99%; background-color: inherit; margin-left: auto; margin-right: auto" | style="background-color: Cornsilk; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" colspan="2" |[[File:Flag_of_the_Philippines.svg|frameless|100x100px]] <big>'''The Philippine Languages Department'''</big> |- | style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}<big>Welcome!</big>{{center bottom}} {{center top}}<big>Maugmang Pag-abot!</big>{{center bottom}} [[File:Satellite image of Philippines in March 2002.jpg|150px|right]] ==Bikol 1== [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]], a foundational course provided by the [[Portal:Southeast Asian languages|Division of Southeast Asian Languages]], serves as an entry point for students seeking to acquaint themselves with the rudimentary aspects of the language, encompassing uncomplicated vocabulary acquisition, the mastering of fundamental phrases, and the exploration of elementary grammatical structures. Throughout this course, participants embark on an immersive linguistic journey, progressively building their proficiency in Filipino through engaging lessons, interactive exercises, and practical conversational practice that empowers them to communicate effectively in real-world scenarios. | rowspan="2" style="width: 40%; background-color: #efefff; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | {{center top}}<h2>Bicol</h2>{{center bottom}} {{center top}} [[File:Mayon Volcano - Camalig.jpg|Mayon Volcano|350px|center]] {{center bottom}} '''Bicol''' is an administrative region of the Philippines located at the southeast end of Luzon island. It is a peninsula with four provinces - Albay, Camarines Norte, Camarines Sur and Sorsogon and two island provinces - Catanduanes and Masbate. It is officially known as Bicol Region and given the designation as Region V. |- | style="width: 60%; background-color: Cornsilk; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" | [[Image:Nuvola apps edu miscellaneous.svg|right|96px]] ==Lessons== * Lesson #1<nowiki>: </nowiki>'''[[/Lesson:Vocabulary|Vocabulary]]''' * Lesson #2<nowiki>: </nowiki>'''[[/Lesson:Grammar|Grammar]]''' |} [[Category:Bikol]] b07ppsg0qev50odlxpsfdgue29c8v6i Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary 0 328925 2803483 2026-04-08T06:28:48Z CarlessParking 3064444 Created page with "Days and Months are borrowed from Spanish {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |Octob..." 2803483 wikitext text/x-wiki Days and Months are borrowed from Spanish {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} 5530hffldiukt2zw96iu0y2z9nntid9 2803486 2803483 2026-04-08T06:45:27Z CarlessParking 3064444 2803486 wikitext text/x-wiki Days and Months are borrowed from Spanish {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} 6i3s4an2s0fqh7xpein1az231g3uou5 2803487 2803486 2026-04-08T06:45:57Z CarlessParking 3064444 2803487 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} 7efueyroik8z256f6qb8jgew2mhxkvn 2803489 2803487 2026-04-08T06:50:22Z CarlessParking 3064444 2803489 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} [[Portal:Southeast Asian languages|Back to main page]]. ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} fxqt0s6aoc984zuraq0nrvwk86qw4wt 2803490 2803489 2026-04-08T06:50:45Z CarlessParking 3064444 2803490 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} 7efueyroik8z256f6qb8jgew2mhxkvn 2803491 2803490 2026-04-08T06:50:57Z CarlessParking 3064444 /* Spatio-temporal Dimensions */ 2803491 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} [[Portal:Southeast Asian languages|Back to main page]]. jjcvax9l5jwojr8d6fy1rohc57znnjd 2803527 2803491 2026-04-08T09:40:21Z CarlessParking 3064444 /* Spatio-temporal Dimensions */ 2803527 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} Here are some common phrases in Bikol. {| border=0 cellpadding=3 cellspacing=0 |- bgcolor=#eeeeee ! align=left | English ! align=left | Bikol ! align=left | Pseudo pronunciation |- |hello||kumusta||''koo-mooss-TAH''|| |- |how are you?||kumusta ka?||''koo-mooss-TAH kah?''|| |- |thank you||Dios mabalos||''JAWSS MAH-bah-lawss''|| |- |you're welcome||daing ano man||''dah-EENG ah-NAW mahn''|| |- |yes||iyo||''ee-YAW''|| |- |no||dai||''dah-EE''|| |- |I am fine||marhay man ako||''mahr-HIGH mahn ah-KAW''|| |- |what is your name?||ano an pangaran mo?||''ah-NAW ahn pah-NGAH-rahn maw?''|| |- |my name is ...||ako si...||''ah-KAW see''|| |- |nice to know you||Kaugmahan ko na mamidbidan ka||''kah-oog-MAH-hahn kaw nah mah-meed-BEE-dahn kah''|| |- |nice to know you too||Kaugmahan ko man na mamidbidan ka||''kah-oog-MAH-hahn kaw mahn nah mah-meed-BEE-dahn kah''|| |- |How much does this cost?||manggurano ini?||''mahng-goo-RAH'-naw ee-NEE?''|| |- |Do you speak English?||Tatao ka mag-Ingles?||''tah-tah-AW kah mahg-eeng-GLEHSS''|| |} [[Portal:Southeast Asian languages|Back to main page]]. 73c8q7awzoxk7rpwa33hgb3poiwxo9g 2803531 2803527 2026-04-08T09:52:59Z CarlessParking 3064444 /* Spatio-temporal Dimensions */ 2803531 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} ==Common phrases== Here are some common phrases in Bikol. {| border=0 cellpadding=3 cellspacing=0 |- bgcolor=#eeeeee ! align=left | English ! align=left | Bikol ! align=left | Pseudo pronunciation |- |hello||kumusta||''koo-mooss-TAH''|| |- |how are you?||kumusta ka?||''koo-mooss-TAH kah?''|| |- |thank you||Dios mabalos||''JAWSS MAH-bah-lawss''|| |- |you're welcome||daing ano man||''dah-EENG ah-NAW mahn''|| |- |yes||iyo||''ee-YAW''|| |- |no||dai||''dah-EE''|| |- |I am fine||marhay man ako||''mahr-HIGH mahn ah-KAW''|| |- |what is your name?||ano an pangaran mo?||''ah-NAW ahn pah-NGAH-rahn maw?''|| |- |my name is ...||ako si...||''ah-KAW see''|| |- |nice to know you||Kaugmahan ko na mamidbidan ka||''kah-oog-MAH-hahn kaw nah mah-meed-BEE-dahn kah''|| |- |nice to know you too||Kaugmahan ko man na mamidbidan ka||''kah-oog-MAH-hahn kaw mahn nah mah-meed-BEE-dahn kah''|| |- |How much does this cost?||manggurano ini?||''mahng-goo-RAH'-naw ee-NEE?''|| |- |Do you speak English?||Tatao ka mag-Ingles?||''tah-tah-AW kah mahg-eeng-GLEHSS''|| |} [[Portal:Southeast Asian languages|Back to main page]]. fkycu8n9jz7f94ndw2dzvk5f9inhdle 2803559 2803531 2026-04-08T11:45:39Z CarlessParking 3064444 /* Common phrases */ 2803559 wikitext text/x-wiki Days and Months are borrowed from Spanish. They are all written beginning with capital letters. {| class="wikitable" |+ !Engling !Bikol |- |Monday |Lunes |- |Tuesday |Martes |- |Wednesday |Miyerkoles |- |Thursday |Huwebes |- |Friday |Biyernes |- |Saturday |Sabado |- |Sunday |Domingo |} {| class="wikitable" |+ !English !Bikol |- |January |Enero |- |February |Pebrero |- |March |Marso |- |April |Abril |- |May |Mayo |- |June |Hunyo |- |July |Hulyo |- |August |Agosto |- |September |Setyembre |- |October |Oktubre |- |November |Nobyembre |- |December |Disyembre |} ==Spatio-temporal Dimensions== Bikol has ha– as prefix for a special class of adjectives, with the rest of adjectives using ‘ma-. Ha– is affixed only to bases indicating spatio-temporal dimensions. {| class="wikitable" |- ! Bikol !! English |- | halangkaw || tall |- | hababâ || low |- | hararom || deep |- | hababaw || shallow |- | halìpot ||short |- | halabà || long (spatial) |- | harayô || far |- | harani || near |- | hayakpit|| narrow |- | halakbang || wide |- | halawig|| long (temporal) |- | haloy || long (temporal) |} ==Common phrases== Here are some common phrases in Bikol. {| border=0 cellpadding=3 cellspacing=0 |- bgcolor=#eeeeee ! align=left | English ! align=left | Bikol ! align=left | Pseudo pronunciation |- |hello||kumusta||''koo-mooss-TAH''|| |- |how are you?||kumusta ka?||''koo-mooss-TAH kah?''|| |- |thank you||Dios mabalos||''JAWSS MAH-bah-lawss''|| |- |you're welcome||daing ano man||''dah-EENG ah-NAW mahn''|| |- |yes||iyo||''ee-YAW''|| |- |no||dai||''dah-EE''|| |- |I am fine||marhay man ako||''mahr-HIGH mahn ah-KAW''|| |- |what is your name?||ano an pangaran mo?||''ah-NAW ahn pah-NGAH-rahn maw?''|| |- |my name is ...||ako si...||''ah-KAW see''|| |- |nice to know you||Kaugmahan ko na mamidbidan ka||''kah-oog-MAH-hahn kaw nah mah-meed-BEE-dahn kah''|| |- |nice to know you too||Kaugmahan ko man na mamidbidan ka||''kah-oog-MAH-hahn kaw mahn nah mah-meed-BEE-dahn kah''|| |- |How much does this cost?||manggurano ini?||''mahng-goo-RAH'-naw ee-NEE?''|| |- |Do you speak English?||Tatao ka mag-Ingles?||''tah-tah-AW kah mahg-eeng-GLEHSS''|| |} [[Portal:Southeast Asian languages|Back to main page]] • [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Grammar|Grammar]] 9vuwrrom8u3sxqmwvukerj5gf4cjdbd Southeast Asian Languages/Philippine Languages/Bikol 1 0 328926 2803497 2026-04-08T07:18:24Z CarlessParking 3064444 Created page with "This is how you introduce yourself in Bikol. ;Kumusta? : How are you? ;Marhay man. : Fine. ;Ako si ____. : My name is ____. (''literally'' I am ____.) ;Anong pangaran mo? : What is your name? [[Portal:Southeast Asian languages|Back to main page]]. [[Category:Bikol]]" 2803497 wikitext text/x-wiki This is how you introduce yourself in Bikol. ;Kumusta? : How are you? ;Marhay man. : Fine. ;Ako si ____. : My name is ____. (''literally'' I am ____.) ;Anong pangaran mo? : What is your name? [[Portal:Southeast Asian languages|Back to main page]]. [[Category:Bikol]] pcuinnv7h0vwpqhmg63oyp8lsshhh8e 2803498 2803497 2026-04-08T07:21:28Z CarlessParking 3064444 2803498 wikitext text/x-wiki This is how you introduce yourself in Bikol. ;Kumusta? : How are you? ;Marhay man. : Fine. ;Ako si ____. : My name is ____. (''literally'' I am ____.) ;Anong pangaran mo? : What is your name?<br> If you want to greet everyone say: ;Marhay na aldaw : Good day ;Marhay na aga : Good morning ;Marhay na udto : Good noon ;Marhay na hapon : Good afternoon ;Marhay na banggi : Good evening [[Portal:Southeast Asian languages|Back to main page]]. [[Category:Bikol]] 47imo7cw8xv73wwg6i0xuyym69bvti2 2803561 2803498 2026-04-08T11:47:20Z CarlessParking 3064444 2803561 wikitext text/x-wiki This is how you introduce yourself in Bikol. ;Kumusta? : How are you? ;Marhay man. : Fine. ;Ako si ____. : My name is ____. (''literally'' I am ____.) ;Anong pangaran mo? : What is your name?<br> If you want to greet everyone say: ;Marhay na aldaw : Good day ;Marhay na aga : Good morning ;Marhay na udto : Good noon ;Marhay na hapon : Good afternoon ;Marhay na banggi : Good evening [[Portal:Southeast Asian languages|Back to main page]] • [[Southeast Asian Languages/Philippine Languages/Bikol 2|Bikol 2]] [[Category:Bikol]] k8sg656xcu1uvmjt3qtmvao6klwuyvi Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Grammar 0 328929 2803533 2026-04-08T10:00:43Z CarlessParking 3064444 Created page with "Word connectives or “ligatures” are a unique part of the Bikol language that are used to link two words together. There are a variety of grammar patterns that require words to be connected by ligatures. The first pattern to learn, is that you should connect nouns and the adjectives that describe them using ligatures. For example, in the phrase: “beautiful maiden”, the words “beautiful” and “maiden” should be connected with a ligature in the Bikol lang..." 2803533 wikitext text/x-wiki Word connectives or “ligatures” are a unique part of the Bikol language that are used to link two words together. There are a variety of grammar patterns that require words to be connected by ligatures. The first pattern to learn, is that you should connect nouns and the adjectives that describe them using ligatures. For example, in the phrase: “beautiful maiden”, the words “beautiful” and “maiden” should be connected with a ligature in the Bikol language. Two types of ligatures: 1.) If the first word ends in a vowel, the ligature -ng is attached to the end of that word to connect it to the next word. Example:<br/> daragang magayon ''(beautiful maiden)'' 2.) If the first word ends in any consonant including the letter "N", the word ''na'' is used to connect two words. Example:<br/> magayon na daraga ''(beautiful maiden)'' o41f0834atulnrul1rb3w19dt57oh9n 2803534 2803533 2026-04-08T10:01:48Z CarlessParking 3064444 2803534 wikitext text/x-wiki Word connectives or “ligatures” are a unique part of the Bikol language that are used to link two words together. There are a variety of grammar patterns that require words to be connected by ligatures. The first pattern to learn, is that you should connect nouns and the adjectives that describe them using ligatures. For example, in the phrase: “beautiful maiden”, the words “beautiful” and “maiden” should be connected with a ligature in the Bikol language. Two types of ligatures: 1.) If the first word ends in a vowel, the ligature -ng is attached to the end of that word to connect it to the next word. Example:<br/> daragang magayon ''(beautiful maiden)'' 2.) If the first word ends in any consonant including the letter "N", the word ''na'' is used to connect two words. Example:<br/> magayon na daraga ''(beautiful maiden)'' [[Portal:Southeast Asian languages|Back to main page]]. 3z35cs76bkhgxhibl0cx5erhwdga91b 2803560 2803534 2026-04-08T11:46:20Z CarlessParking 3064444 2803560 wikitext text/x-wiki Word connectives or “ligatures” are a unique part of the Bikol language that are used to link two words together. There are a variety of grammar patterns that require words to be connected by ligatures. The first pattern to learn, is that you should connect nouns and the adjectives that describe them using ligatures. For example, in the phrase: “beautiful maiden”, the words “beautiful” and “maiden” should be connected with a ligature in the Bikol language. Two types of ligatures: 1.) If the first word ends in a vowel, the ligature -ng is attached to the end of that word to connect it to the next word. Example:<br/> daragang magayon ''(beautiful maiden)'' 2.) If the first word ends in any consonant including the letter "N", the word ''na'' is used to connect two words. Example:<br/> magayon na daraga ''(beautiful maiden)'' [[Portal:Southeast Asian languages|Back to main page]] • [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] mjfndmx2yzn3zk7ht7g86ms3to86fzg Complex Analysis/decomposition theorem 0 328930 2803544 2026-04-08T10:37:08Z Bert Niehaus 2387134 Bert Niehaus moved page [[Complex Analysis/decomposition theorem]] to [[Complex Analysis/Decomposition theorem]]: capitalize first character 2803544 wikitext text/x-wiki #REDIRECT [[Complex Analysis/Decomposition theorem]] 101ybp0yes3e3zkaafgrxk5w0vj9jrs Complex Analysis/Residuals 0 328931 2803547 2026-04-08T10:38:17Z Bert Niehaus 2387134 Bert Niehaus moved page [[Complex Analysis/Residuals]] to [[Complex Analysis/Residue]]: Correct title of learning resource 2803547 wikitext text/x-wiki #REDIRECT [[Complex Analysis/Residue]] 2ohf4520iay5b93ccs708wc1bkyj7uo Riemann Removability Theorem 0 328932 2803553 2026-04-08T10:49:16Z Bert Niehaus 2387134 Bert Niehaus moved page [[Riemann Removability Theorem]] to [[Riemann's theorem on removable singularities]]: Riemann's theorem on removable singularities 2803553 wikitext text/x-wiki #REDIRECT [[Riemann's theorem on removable singularities]] ssk4bszgaw4uze8nksts27jezvdc7ks Category:Wikipedia books 14 328933 2803556 2026-04-08T11:21:20Z MarioeMary 3063819 Created page with "{{Category redirect|Wikiversity books}} {{Wikipedia|simple:Category:Wikipedia books}} {{Commons|Category:Wikipedia books}}" 2803556 wikitext text/x-wiki {{Category redirect|Wikiversity books}} {{Wikipedia|simple:Category:Wikipedia books}} {{Commons|Category:Wikipedia books}} sgh6ipy7zg5xr6awx76bk0ylwp4sugv Southeast Asian Languages/Philippine Languages/Bikol 2 0 328934 2803562 2026-04-08T11:50:08Z CarlessParking 3064444 Created page with " {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |A a |{{IPA|a}} |''far'' |- |E e |{{IPA|e}} |''hen'' |- |I i |{{IPA|i}} |''see'' |- |O o |{{IPA|o}} |''order'' |- |U u |{{IPA|u}} |''soon'' |}" 2803562 wikitext text/x-wiki {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |A a |{{IPA|a}} |''far'' |- |E e |{{IPA|e}} |''hen'' |- |I i |{{IPA|i}} |''see'' |- |O o |{{IPA|o}} |''order'' |- |U u |{{IPA|u}} |''soon'' |} k5wiwti8sfusngd5i306dofwjfqjdmt 2803563 2803562 2026-04-08T11:52:32Z CarlessParking 3064444 2803563 wikitext text/x-wiki {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |A a |{{IPA|a}} |''far'' |- |E e |{{IPA|e}} |''hen'' |- |I i |{{IPA|i}} |''see'' |- |O o |{{IPA|o}} |''order'' |- |U u |{{IPA|u}} |''soon'' |} |Q q |{{IPA|kw}} |''qu''est (always with a silent '''''u''''') |- |V v |{{IPA|b}} |pronounced the same way as '''''b''''' (see above) like '''''b''''' in '''''b'''''ow. |- |X x |{{IPA|ks}} |like '''''x''''' in fle'''''x'''''ible :like '''''ss''''' in hi'''''ss''''' (at beginning of a word) :like '''''h''''' as in '''''h'''''e in the family name ''Roxas'' |- |Z z |{{IPA|s}} | '''''s''''' in '''''s'''''upper |} 2xjz0iknq3iqb1jhbscxlylnsbtcasc 2803564 2803563 2026-04-08T11:55:13Z CarlessParking 3064444 2803564 wikitext text/x-wiki {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |A a |{{IPA|a}} |''far'' |- |E e |{{IPA|e}} |''hen'' |- |I i |{{IPA|i}} |''see'' |- |O o |{{IPA|o}} |''order'' |- |U u |{{IPA|u}} |''soon'' |} {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |B b |{{IPA|b}} |''bee'' |- |K k |{{IPA|k}} |''key'' |- |D d |{{IPA|d}} |''dough'' |- |G g |{{IPA|g}} |''gold'' |- |H h |{{IPA|h}} |''heat'' |- |L l |{{IPA|l}} |''let'' |- |M m |{{IPA|m}} |''me'' |- |N n |{{IPA|n}} |''nice'' |- |NG ng |{{IPA|ŋ}} |''song'' |- |P p |{{IPA|p}} |''pea'' |- |R r |{{IPA|r}} |''raw'' |- |S s |{{IPA|s}} |''sea'' |- |T t |{{IPA|t}} |''tea'' |- |W w |{{IPA|w}} |''weak'' |- |Y y |{{IPA|y}} |''you'' |} Some consonants are borrowed from Spanish and English and are used in writing names of places and personal names. {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |C c |{{IPA|k, s}} |'''''c'''''all, '''''c'''''orn, '''''c'''''ure (before '''''a''''', '''''o''''', '''''u''''') :like '''''c''''' in '''''c'''''ell, '''''c'''''inder (before '''''e''''' or '''''i''''') |- |F f |{{IPA|f}} |'''''f'''''ine |- |J j |{{IPA|h}} |'''''h''''' in '''''h'''''am |- |LL ll |{{IPA|lj}} |m''ll''ion |- |Ñ ñ |{{IPA|nj}} |ca''ny''on |- |Q q |{{IPA|kw}} |''qu''est (always with a silent '''''u''''') |- |V v |{{IPA|b}} |pronounced the same way as '''''b''''' (see above) like '''''b''''' in '''''b'''''ow. |- |X x |{{IPA|ks}} |like '''''x''''' in fle'''''x'''''ible :like '''''ss''''' in hi'''''ss''''' (at beginning of a word) :like '''''h''''' as in '''''h'''''e in the family name ''Roxas'' |- |Z z |{{IPA|s}} | '''''s''''' in '''''s'''''upper |} 88fbhmartaett6ro972y4z9krri31pq 2803565 2803564 2026-04-08T11:57:57Z CarlessParking 3064444 2803565 wikitext text/x-wiki *'''Vowels''' **''Upper case'': A E I O U **''Lower case'': a e i o u *'''Original consonants''' **''Upper case'': B K D G H L M N NG P R S T W Y **''Lower case'': b k d g h l m n ng p r s t w y *'''Additional consonants''' **''Upper case'': C F J Ñ Q V X Z **''Lower case'': c f j ñ q v x z {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |A a |{{IPA|a}} |''far'' |- |E e |{{IPA|e}} |''hen'' |- |I i |{{IPA|i}} |''see'' |- |O o |{{IPA|o}} |''order'' |- |U u |{{IPA|u}} |''soon'' |} {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |B b |{{IPA|b}} |''bee'' |- |K k |{{IPA|k}} |''key'' |- |D d |{{IPA|d}} |''dough'' |- |G g |{{IPA|g}} |''gold'' |- |H h |{{IPA|h}} |''heat'' |- |L l |{{IPA|l}} |''let'' |- |M m |{{IPA|m}} |''me'' |- |N n |{{IPA|n}} |''nice'' |- |NG ng |{{IPA|ŋ}} |''song'' |- |P p |{{IPA|p}} |''pea'' |- |R r |{{IPA|r}} |''raw'' |- |S s |{{IPA|s}} |''sea'' |- |T t |{{IPA|t}} |''tea'' |- |W w |{{IPA|w}} |''weak'' |- |Y y |{{IPA|y}} |''you'' |} Some consonants are borrowed from Spanish and English and are used in writing names of places and personal names. {| class="wikitable" |- ! Letter ! IPA ! Pronunciation of the letter (English approximation) |- |C c |{{IPA|k, s}} |'''''c'''''all, '''''c'''''orn, '''''c'''''ure (before '''''a''''', '''''o''''', '''''u''''') :like '''''c''''' in '''''c'''''ell, '''''c'''''inder (before '''''e''''' or '''''i''''') |- |F f |{{IPA|f}} |'''''f'''''ine |- |J j |{{IPA|h}} |'''''h''''' in '''''h'''''am |- |LL ll |{{IPA|lj}} |m''ll''ion |- |Ñ ñ |{{IPA|nj}} |ca''ny''on |- |Q q |{{IPA|kw}} |''qu''est (always with a silent '''''u''''') |- |V v |{{IPA|b}} |pronounced the same way as '''''b''''' (see above) like '''''b''''' in '''''b'''''ow. |- |X x |{{IPA|ks}} |like '''''x''''' in fle'''''x'''''ible :like '''''ss''''' in hi'''''ss''''' (at beginning of a word) :like '''''h''''' as in '''''h'''''e in the family name ''Roxas'' |- |Z z |{{IPA|s}} | '''''s''''' in '''''s'''''upper |} 582iie895hxjuwwqjeerymoinm6vq5i